Impact ofWater-Level Variations999168/... · 2016-09-30 · LICENTITE TH ESIS Department of Civil, Environmental and Natural Resources Engineering1 Division of Mining and Geotechnical

LICENTIATE T H E S I S

Department of Civil, Environmental and Natural Resources Engineering1Division of Mining and Geotechnical Engineering

Impact of Water-Level Variations on Slope Stability

Jens Johansson

ISSN 1402-1757ISBN 978-91-7439-958-5 (print)ISBN 978-91-7439-959-2 (pdf)

Luleå University of Technology 2014

Jens Johansson Impact of W

ater-Level Variations on Slope Stability

ISSN: 1402-1757 ISBN 978-91-7439-XXX-X Se i listan och fyll i siffror där kryssen är

LICENTIATE THESIS

Impact of Water-Level Variations on Slope Stability

Jens M. A. Johansson

Luleå University of Technology Department of Civil, Environmental and Natural Resources Engineering

Division of Mining and Geotechnical Engineering

Printed by Luleå University of Technology, Graphic Production 2014

ISSN 1402-1757 ISBN 978-91-7439-958-5 (print)ISBN 978-91-7439-959-2 (pdf)

Luleå 2014

www.ltu.se

Preface

i

PREFACE

This work has been carried out at the Division of Mining and Geotechnical

Engineering at the Department of Civil, Environmental and Natural Resources, at

Luleå University of Technology. The research has been supported by the Swedish

Hydropower Centre, SVC; established by the Swedish Energy Agency, Elforsk and

Svenska Kraftnät together with Luleå University of Technology, The Royal Institute

of Technology, Chalmers University of Technology and Uppsala University.

I would like to thank Professor Sven Knutsson and Dr. Tommy Edeskär for their

support and supervision.

I also want to thank all my colleagues and friends at the university for contributing to

pleasant working days.

Jens Johansson, June 2014

Impact of water-level variations on slope stability

ii

Abstract

iii

ABSTRACT

Waterfront-soil slopes are exposed to water-level fluctuations originating from either

natural sources, e.g. extreme weather and tides, or from human activities such as

watercourse regulation for irrigation, freshwater provision, hydropower production

etc. Slope failures and bank erosion is potentially getting trees and other vegetation

released along with bank landslides. When floating debris is reaching hydropower

stations, there will be immediate risks of adverse loading on constructions, and

clogging of spillways; issues directly connected to as well energy production as dam

safety.

The stability of a soil slope is governed by slope geometries, stress conditions, and soil

properties. External water loading, pore-pressure changes, and hydrodynamic impact

from water flow are factors being either influencing, or completely governing the

actual soil properties. As a part of this study, knowledge concerning water-level

fluctuations has been reviewed; sources, geotechnical effects on slopes, and approaches

used for modelling, have been focused. It has been found a predominance of research

focused on coastal erosion, quantification of sediment production, bio-environmentally

issues connected to flooding, and effects on embankment dams subjected to rapid

drawdown. Though, also water-level rise has been shown to significantly influence

slope stability. There seems to be a need for further investigations concerning effects of

rapidly increased water pressures, loss of negative pore pressures, retrogressive failure

development, and long-term effects of recurring rise-drawdown cycling.

Transient water flow within soil structures affects pore-pressure conditions, strength,

and deformation behavior of the soil. This in turn does potentially lead to soil-material

migration, i.e. erosion. This process is typically considered in the context of

embankment dams. Despite the effects of transient water flow, the use of simple limit-

equilibrium methods for slope analysis is still widely spread. Though, improved

accessibility of high computer capacity allows for more and more advanced analyses to

be carried out. In addition, optimized designs and constructions are increasingly

demanded, meaning less conservative design approaches being desired. This is not at

least linked to economic as environmental aspects. One non-conservative view of

slope-stability analysis regards consideration of negative pore pressures in unsaturated

soils. In this study, three different approaches used for hydro-mechanical coupling in

FEM-modelling of slope stability, were evaluated. A fictive slope consisting of a well-

graded postglacial till was exposed to a series of water-level fluctuation cycles.

Modelling based on classical theories of dry/fully saturated soil conditions, was put

against two more advanced approaches with unsaturated-soil behavior considered. In


iv

the classical modelling, computations of pore-pressures and deformations were run

separately, whereas the advanced approaches did allow for computations of pore-

pressures and deformation to be fully coupled. The evaluation was carried out by

comparing results concerning stability, vertical displacements, pore pressures, flow, and

model-parameter influence.

It was found that the more advanced approaches used did capture variations of pore

pressures and flow to a higher degree than did the classical, more simple approach.

Classical modelling resulted in smaller vertical displacements and smoother pore-

pressure and flow developments. Flow patterns, changes of soil density governed by

suction fluctuations, and changes of hydraulic conductivity, are all factors governing as

well water-transport (e.g. dissipation of excess pore pressures) as soil-material transport

(e.g. susceptibility to internal erosion to be initiated and/or continued). Therefore, the

results obtained underline the strengths of sophisticated modelling.

Abbreviations and symbols

v

ABBREVIATIONS AND SYMBOLS

Symbol/

Abbreviation Definition Unit

A1 Modelling approach 1 -



Cohesion ( ’ for drained conditions) kPa

Soil weight kN/m3

Saturated soil weight kN/m3

Shear strain -

Unsaturated soil weight kN/m3

Strain increment (major principal-stress direction) -

Volumetric strain increment -

Strain increment (x-direction) -

Elastic strain increment (x-direction) -

Plastic strain increment (x-direction) -

Volume change m3

, Displacements (x- and y-direction) m

Initial lengths (x- and y-direction) m

Head loss m

Flow length m

Horizontal side force kN

Horizontal side force, adjacent slice kN

Modulus, initial stiffness kPa

Modulus, secant (50% of deviatoric peak stress) kPa

Oedometer modulus kPa

Modulus, unloading/reloading kPa

Young´s modulus kPa

External water level -

Void ratio -

Deviatoric strain (engineering notation) -

Axial strain -

Deviatoric strain (triaxial shear strain) -

, Normal strains (x- and y-direction) -

Principal strains -

Volumetric strain -

Elastic strain -

Plastic strain -

Plastic strain rate 1/s


vi

Visco-plastic strain rate 1/s

FE(M) Finite element (method) -

FOS Factor of safety -

( ) Yield function -

Shear modulus kPa

Ground surface -

Gruondwater table -

Acceleration of gravity m/s2

( ) Plastic potential function -

, , van Genuchten´s empirical parameters -

Suction pore-pressure head m

Sensitivity ratio -

Snesitivity score -

Hydraulic gradient -

Bulk modulus kPa

Water bulk modulus kPa

Hydraulic conductivity m/s

Saturated hydraulic conductivity m/s

Relative hydraulic conductivity -

Intrinsic hydraulic conductivity m2

Length of slice base m

LE(M) Limit-equilibrium (method) -

Dynamic viscosity kPa s

Normal force N

Porosity -

Poisson´s ratio -

Specific volume -

Effective normal force at slide base N

PSH Pumped-storage hydropower -

Gradient of the pore pressure kPa/m

Preconsolidation pressure kPa

Effective mean stress kPa

Deviatoric stress kPa

Specific discharge m/s

Horizontal flow m/day

Vertical flow m/day

Vertical flow, upward directed m/day

Vertical flow, downward directed m/day

Maximum flow m/day

Abbreviations and symbols

vii

Horizontal flow, rightward directed m/day

Particle roundness -

Rapid drawdown -

Radius of slip surface (Swedish method) m

Initial radius of slip surface (log-spiral method) m

Water density kg/m3

Shear force at slice base N

Slip surface -

Degree of saturation -

Effective degree of saturation -

Maximum degree of saturation -

Minimum degree of saturation -

Residual degree of saturation -

Saturated degree of saturation -

Deviatoric stresses components kPa

Total normal stress kPa

Effective normal stress kPa

Axial effective stress kPa

Viscous stress component kPa

Inviscid stress component kPa

Major principal effective stress kPa

Horizontal shear force N

Shear stress kPa

Maximum shear stress kPa

Shear strength kPa

Water-pressure force N

Pore pressure kPa

Pore-air pressure kPa

Pore-water pressure kPa

Vertical deformation m

Total volume m3

Pore volume m3

Volume of solids m3

Water volume m3

Water flow rate m/s

Slice weight N

Water load -

Water-level fluctuation cycle -

Angle of friction ( for drained conditions)


viii

Mobilized friction angle

Friction angle at critical state Dilation component of the friction angle Geometrical interference component of friction angle

Particle-pushing component of friction angle

Inter-particle component of the friction angle

Vertical side force N

Vertical side force, adjacent slice N

Bishop´s parameter -

Dilatancy angle Gradient operator -

Table of Contents

ix

TABLE OF CONTENTS

PREFACE .............................................................................................. I

ABSTRACT ......................................................................................... III

ABBREVIATIONS AND SYMBOLS .................................................. V

TABLE OF CONTENTS ................................................................... IX

1 INTRODUCTION ....................................................................... 1

1.1 Background ............................................................................ 1 1.2 Objectives .............................................................................. 4

1.2.1 Aim and objectives ....................................................... 4

1.2.2 Method ........................................................................ 5

1.3 Scope and delimitations .......................................................... 5 1.4 Outline ................................................................................... 6

2 WATER-LEVEL VARIATION – SOURCES, EFFECTS, AND CONSIDERATION ...................................................................... 7

2.1 Sources ................................................................................... 7

2.1.1 Long-term water-level variations .................................. 7

2.1.2 Short-term water-level variations ................................. 7 2.2 Effects ................................................................................... 10

2.3 Consideration ....................................................................... 11

3 SLOPE ANALYSIS ...................................................................... 15 3.1 Introduction ......................................................................... 15

3.2 Description and classification of soil ...................................... 15

3.3 Stress and strength ................................................................ 16 3.3.1 Stress .......................................................................... 16

3.3.2 Particle arrangement................................................... 17

3.3.3 Strength and failure .................................................... 18 3.3.4 Angle of internal friction ............................................ 19

3.3.5 Factors governing internal friction –

changed properties ..................................................... 19

3.4 Slope-stability analysis ........................................................... 22 3.4.1 Introduction ............................................................... 22

3.4.2 Limit-equilibrium analysis ......................................... 23

3.4.3 Deformation analysis .................................................. 25

3.4.4 Drainage conditions ................................................... 26

4 MODELLING OF SOIL AND WATER .................................... 29

4.1 Constitutive description of soil ............................................. 29 4.1.1 Stress .......................................................................... 29


x

4.1.2 Strain ......................................................................... 29 4.1.3 Elastic response .......................................................... 30

4.1.4 Unloading/reloading .................................................. 32

4.1.5 Plastic response .......................................................... 32

4.1.6 Mohr-Coulomb model .............................................. 35 4.2 Hydraulic modelling ............................................................. 38

4.2.1 Fundamentals ............................................................. 38

4.2.2 Unsaturated soil ......................................................... 38

4.2.3 Flow .......................................................................... 39 4.2.4 van Genuchten model ................................................ 40

4.3 Parameter sensitivity ............................................................. 41

5 FEM-MODELLING .................................................................... 43 5.1 Main aim .............................................................................. 43

5.2 Strategy ................................................................................ 43

5.3 Geometry, materials, and definition of changes ..................... 44 5.4 Models used ......................................................................... 45

6 RESULTS AND COMMENTS .................................................. 47 6.1 Evaluation of FEM-modelling approaches ............................ 47

6.1.1 General ...................................................................... 47

6.1.2 Stability ...................................................................... 47

6.1.3 Deformations, pore pressures and flow ....................... 48 6.2 Parameter influence .............................................................. 53

7 DISCUSSION .............................................................................. 57

8 CONCLUSIONS ........................................................................ 61

8.1 General................................................................................. 61 8.2 To be further considered ...................................................... 62

REFERENCES .................................................................................... 65

APPENDED PAPERS:

J. M. Johansson and T. Edeskär, “Effects of External Water-Level Fluctuations

on Slope Stability,” Electron. J. Geotech. Eng., vol. 19, no. K, pp. 2437–2463,

2014.

J. M. Johansson and T. Edeskär, “Modelling approaches considering impacts of

water-level fluctuations on slope stability” To be submitted.

Introduction

1

1 INTRODUCTION

1.1 Background

Regarding the fact that an external water pressure acts stabilizing to a

slope or to an embankment dam: “This is perhaps the o ly good thi g that

water a do to a slope” (J. M. Duncan & Wright, 2005)

There is a worldwide increasing need of land-use in costal/waterfront areas (e.g.

Singhroy, 1995 among others). This trend, together with continuously changing site-

specific conditions, brings more and more advanced and challenging geotechnical

issues. It is necessary to monitor sites, to properly analyze data, and to evaluate

potential risks concerning property, environmental, and human values. In case of

unstable watercourse bank slopes, trees and other vegetation are potentially released

along with bank landslides. In situations where the material released—floating debris—

reaches downstream hydropower stations, there will be potential risks of damaging

loads on constructions and clogging of spillways (e.g. Minarski, 2008; Åstrand &

Johansson, 2011).

The stability of a slope is utterly governed by soil properties, stress conditions, and

slope geometries. Any change taking place of at least one of these factors, means slope-

stability conditions being potentially affected. When it comes to changed soil

properties, there are different scales. At a micro scale, the inherent properties of a soil

are governed by its history; no matter if the soil is processed (crushed, filled etc.), or if

it is naturally occurring; i.e. formed by weathering of rock, transported by erosive

processes, and finally deposited from water, wind, or ice. Also at a larger scale—

considering the soil skeleton—many different processes are governing the properties of

the soil; e.g. particle-size distribution, soil-profile homogeneity, denseness etc. The

properties of a soil is continuously affected by long-term processes, including e.g.

transport and depositing (i.e. erosion and land-form development), and aging (i.e.

weathering or other changed chemical or physical conditions). Any soil volume is

continuously affected by hydrological conditions prevailing; present water is either

influencing or completely governing the actual soil properties. At the scale of bank

slopes and embankment dams, the structures are influenced by external water loads,

development of pore pressures, and hydrodynamic impact from internal and external

water flow. In Figure 1-1, some basic modes of water-level changes are defined; (A)


2

external erosion due to small waves or streaming water, (B) water-level drawdown, (C)

water-level rise, and (D) water-level fluctuations.

Figure 1-1: Basic modes of water level change; streaming water (A), water level drawdown (B), raised

water level (C), and fluctuating water level (D). Water loads (WL), positions of the ground-water table (GWT), and the external water level (EWL), are shown.

Sources of water-level fluctuations (WLF’s) may e.g. include tidal water-level

variations (e.g. Ward, 1945; Li, Barry, & Pattiaratchi, 1997; Raubenheimer, Guza, &

Elgar, 1999), variations caused by wind waves (e.g. Bakhtyar, Barry, Li, Jeng, &

Yeganeh-Bakhtiary, 2009), variations caused by other weather-related events (as heavy

rainstorms and/or snow melting), and combinations of various phenomena (e.g.

Zhang, 2013). Natural phenomena do also include time-dependent soil degradation in

terms of e.g. weathering and structural changes. In addition, the stability of waterfront

slopes is influenced by processes caused and driven by human activities. One such

activity is regulation of watercourses, undertaken for water storage, enabling irrigation,

freshwater provision, and/or hydropower production (e.g. Mill et al., 2010; Solvang,

Harby, & Killingtveit, 2012). Despite the fact that regulation patterns seems to be

critically connected to the stability of the reservoir banks, there is a clearly seen

predominance of studies focusing on bio-environmental issues; i.e. endangered habitats

Introduction

3

of plant and animal species. Moreover, among studies addressing geotechnical aspects

of WLF’s, a large proportion have been directed to tidally driven issues; mostly aimed

to describe relationships between an external sea-water level and groundwater level

motions within the adjacent beach slope (e.g. Emery & Foster, 1948; Parlange et al.,

1984; Nielsen, 1990; Thomas, Eldho, & Rastogi, 2013). Others have been addressing

sediment-loading problems; river sediment budgets, changed river shapes caused by

sedimentation etc. (e.g. Darby et al., 2007; Fox et al., 2007; R. Grove, Croke, &

Thompson, 2013). Concerning WLF-effects specifically on slope-stability, the process

of rapid drawdown is described and investigated by many authors (e.g. Lane & Griffiths,

2000; J. M. Duncan & Wright, 2005; Yang et al., 2010; Pinyol, Alonso, Corominas,

& Moya, 2011; López-Acosta et al., 2013). Pore-pressure gaps and increased

hydrological gradients are potentially occurring when water-level changes are rapidly

coming about. Also water-level rise might cause problems, e.g. related to stress

redistributions due to external loading, wetting induced issues such as loss of negative

pore pressures, and seepage effects. Potential consequences might be loss of shear

strength, soil structure collapse, and development of settlements and/or slope failure

(Jia et al., 2009). Despite this fact, there are significantly less studies carried out on this

topic.

Quantification of slope stability is generally about two parts connected; (1) calculation

of the factor of safety (FOS), and (2) determination of the location of the most critical slip

surface. The development of methods for slope-stability analysis was initially about

quantifications of clay behavior; this in the middle of the 19th century (Ward, 1945).

The approach of assuming circular slip surfaces was introduced in 1916 and presented

as a fully described method in the beginning of the 1920s (Krahn, 2003; J. M. Duncan

& Wright, 2005). Numerous approaches for quantification of slope stability by using

equilibrium equations have been presented. Such limit-equilibrium (LE) methods have

been used for a long time (J. M. Duncan & Wright, 1980; Yu et al., 1998; Zheng et

al., 2009) and have been more or less unchanged for decades (Lane & Griffiths, 1999).

Despite well-known limitations of LE-analysis methods (e.g. Ward, 1945; Lane &

Griffiths, 1999; Krahn, 2003) they have been widely used; largely due to their

simplicity and usability. In parallel to the use of LE-approaches, application of

continuum theories and material models has been brining use of methods based on

deformations analysis, often numerically handled using finite-element (FE) analysis.

Improved accessibility of high computer capacity allows for more and more advanced

analyses to be performed. Optimized designs and constructions are increasingly

demanded and less conservative design approaches are therefore often desired. This is

not at least linked to economic and environmental aspects. One non-conservative view


4

in slope-stability analysis regards consideration of negative pore pressures in unsaturated

soils. Taking into account negative pore pressures is generally associated with counting

on extra contributions to the shear strength of the soil, resulting in extra slope stability.

Though, there are still many unanswered questions when it comes to effects of

considering peculiarities of unsaturated soil behavior (Sheng et al., 2013).

Due to a worldwide expansion of watercourse regulation, with operational patterns

directly governed by activities of energy balancing and freshwater provision, it is

important to find reliable methods for analysis of stability effects on areas being exposed

to the regulation.

1.2 Objectives

1.2.1 Aim and objectives

The aim of this study is to identify and enlighten potential impacts on waterfront slopes

subjected to water-level fluctuations, including evaluation of methods for slope-

stability analysis.

1. Provide a review of the overall topic, capturing:

Sources of water-level fluctuations and known effects on slope stability

Fundamentals of mechanical properties of coarse grained soils; this with an

emphasis on changes properties.

Advantages and disadvantages of methods available for slope-stability analysis

(applicable to problems focused in this study).

2. Investigate potential impacts on a slope subjected to water-level fluctuations;

this by FEM-modelling.

3. Evaluate approaches for FEM-modelling of impacts on a slope subjected to

water-level fluctuations.

4. Evaluate potential benefits provided by parameter-influence analysis carried out

in modelling work.

Introduction

5

1.2.2 Method

The objectives were fulfilled by:

1. Conducting a literature review covering the topic of slope stability connected to

water-level fluctuations.

2. Undertaking FEM-modelling for analysis of a slope subjected to water-level

fluctuations; evaluating development of stability/safety, pore pressures, flow and

deformations.

3. Performing a comparative study of different approaches used in FEM-modelling

of a slope subjected to water-level fluctuations.

4. Performing a sensitivity analysis on the parameters used in the modelling carried

out in (2) and (3).

1.3 Scope and delimitations

The present study makes its starting point considering hydropower-production

systems; this regards as well consequences as sources of issues. Though, both geotechnical aspects on the underlying processes and potential values of the findings are

meant to be generally applicable and possibly extended in various directions. Some

delimitations were done:

Although as well the review part as the modelling part is primarily approached

from a perspective of watercourse regulation, also other sources of water-level

fluctuations are considered.

The issues covered in the present work are approached focused on non-

cohesive and low-cohesive embankment materials.

In the FEM-modelling work done, some factors/conditions have been kept

constant:

- The soil material used is defined aimed to have properties being representative for glacial tills with an amount of fines in the lower range.

- 10 water-level fluctuation cycles are considered

- Only one rate is used for definition of water-level changes

- Only one frequency is used for definition of water-level changes

- The water-level fluctuation cycles are run without pauses. Consequently, excess pore pressures potentially developed are not necessarily dissipated.


6

This allows for soil conditions to be varying by an increased number of water-level fluctuations run, and also for secondary effects linked to these

non-constant conditions to be captured.

The modelling part is based on a fictive case, i.e. focused on relative changes and differences. Though, input data for the soil model and the hydraulic model

are based on values within ranges found representative. Since the result

evaluation is comparative the exact values of the parameters chosen are not

critical.

1.4 Outline

The introduction (chapter 1) covers the overall background and objectives of the

project. In chapter 2 the essence of the literature review regarding water-level

fluctuations; sources, potential effects on slope stability, and considerations taken, is

found (fully presented in Paper A). Chapter 3 covers some fundamentals on soil

mechanics and slope-stability analysis. In chapter 4 basics of constitutive description of

soils are presented along with details concerning the models used (in Paper B) for

description of as well soil behavior as hydraulic properties. In chapter 5 the modelling

work is described, and in chapter 6 the results are compiled. An overall discussion and

conclusions drawn are presented in chapter 7 and chapter 8, respectively. The

results/outcome from the review work is presented in chapter 7.

Paper A: A review concerning water-level fluctuations; sources, potential geotechnical

effects on slope stability, and considerations taken.

Paper B: A comparative study evaluating different approaches of hydro-mechanical

coupling in FEM-modelling of impacts on a slope subjected to water-level

fluctuations.

Besides what is presented in the present study, involvement in research on

quantification of soil-particle shape has taken place. Work on implementation of 2D-

image analysis on shape quantification was presented in (Rodriguez, Johansson, &

Edeskär, 2012).

Water-level variation – sources, effects, and consideration

7

2 WATER-LEVEL VARIATION – SOURCES, EFFECTS, AND CONSIDERATION

2.1 Sources

2.1.1 Long-term water-level variations

Sea-level rise is one of the most highlighted and debated direct effect of climate

changes. The Bruun-theory (Bruun, 1988) is a generally accepted mathematical model

developed for quantification of relationships between sea-level rise and coastal erosion.

The model is assuming a closed material balance system between the shore and the

offshore bottom profile. This kind of process is actually about the water-soil interface

itself—not necessarily driven by water level changes—but could also be a consequence

of water motion along the course; i.e. streaming/flowing. Though, a varied mean-level

brings changed water-level ranges within which the short-term variations are

occurring. In this sense, a rising sea level acts as an enabler of erosion (K. Zhang et al.,

2004), whereupon consideration also of long-term changes are of importance.

Hupp (1992) presented a channel-evolution cycle in a six-stage model describing the

development from stable conditions, via some time limited landform changing stages,

ending up with another stable state (Figure 2-1).

These examples of long-term effects—as well coastal erosion as channel evolution—are

showing clear consequences of water-soil interaction, entailing obvious effects on

beaches/banks. Though, henceforth short-term water level variations will be focused.

2.1.2 Short-term water-level variations

Energy provision – water storage

In the 1970s, the energy interest grew significantly due to uncertainties related to

provision of oil. This was partly due to a generally spread reluctance to become

dependent on other countries, partly due to concerns about the total global oil reserve

Bardi (2009). The use of energy is growing worldwide; the rate of increased use and

production is high as well in less developed countries, as in industrialized regions.


8

Moreover, the use of renewable unregulated energy sources (non-fossil sources,

naturally regenerated) is strongly growing. In 2011, as much as 19.0% out of the total

global energy consumption, came from renewable energy sources (to be compared

with 16.7% for 2010) (REN21, 2013). The significant growth of energy sources being

totally dependent on the meteorological situation at a specific time, will increase the

need for regulated (balanced) power (Catrinu, Solvang, & Korpås, 2010; Whittingham,

2008; Connolly, 2010). The concept is about storing energy being produced during

periods when production of wind power (or other non-regulated sources) is exceeding

the demand, and then letting the stored energy being discharged during periods of

higher demands (Figure 2-2). The development and exploitation of non-regulated

energy sources being planned, and the subsequent need of hydropower balancing, will

necessarily require that flow magnitudes and water level heights, to a greater extent

than until present, will vary in the future (Svenska Kraftnät, 2008). In addition to

conventional hydropower, also pumped-storage hydropower (PSH) is a technique

continuously growing. The system consists of two reservoirs of different elevations.

During off-peak electrical demand, water is pumped from a lower reservoir to a higher

one; during on-peak demand, when the energy need is increasing, the water is

discharged and energy is produced (C. Liu et al., 2010). The technology is well-

established (Connolly, 2010), and is currently the only commercially proven large scale

(>100MW) energy storage technology. In Rognlien (2012) some site-specific scenarios

Figure 2-1: Co eptual model o ha el evolutio . The illustrated y le i ludes the stages “pre-

modi ied”, bei g o stable ature (Stage I), ollowed by stages bei g lasti g or limited time periods, i ludi g “ o stru tio ” (Stage II), “degradatio ” (Stage III), “threshold” (Stage IV), a d “aggradation” (Stage V). The y le e ds up with a stage o “re overy”, whi h o e is agai stable (Stage VI). The arrows are representing the movement direction of the bed (vertically) and the banks (horizontally). The pictures are not to scale. (after Hupp, 1992)


9

of practical implementation of PSH were discussed; examples included daily water-

level variations in the order of 10 m. In 2011, there were about 170 pumped-storage

plants operating in Europe and more than 50 new PSH-projects were expected to be

either under construction or planned until 2020 (Zuber, 2011).

Other sources

Natural water-level variations are related to meteorological and geological phenomena.

Tide induced water-level fluctuations have been specifically studied for a long time.

These kinds of variations are important to beach-sediment transport (e.g. Grant, 1948;

J. R. Duncan, 1964; Nielsen, 1990; Li, Barry, Parlange, et al., 1997; Li, Barry, &

Pattiaratchi, 1997). Besides actual sea-level variations and the mechanical material-

transport processes, also the groundwater table on land is affected by the tidal changes

(Emery & Foster, 1948; Chappell, Eliot, Bradshaw, & Lonsdale, 1979). Except tidally

driven variations, there is also wind (i.e. strong winds, hurricanes etc.), changed

atmospheric pressure (causing water movement), and submarine earthquakes

(generating waves sometimes propagating landward) (Pugh, 1987).

Figure 2-2: Schematic sketch of the principle of storing energy produced during a period when production from wind power (or other non-regulated sources) is exceeding the demand, whereupon the stored energy is used during periods of higher demands. (Connolly, 2010)


10

2.2 Effects

Drawdown of an external water level is usually impairing the stability of a slope. The

process rapid drawdown is described and investigated by many authors (e.g. Lane &

Griffiths, 2000; J. M. Duncan & Wright, 2005; Yang et al., 2010; Pinyol, Alonso,

Corominas, & Moya, 2011). Pore pressure gaps and increased hydraulic gradients are

potentially occurring as a consequence of rapid water-level changes. Such pore-

pressure gaps combined with a decreased or fully vanished supporting water load, may

lead to reduced slope stability. Studies on this phenomenon have been focused mostly

on earth-fill dams, rather than on waterfront slopes in general. This predominance

notwithstanding, the connection between rapid drawdown, vertical infiltration, and

slope stability was generally highlighted in Yang et al. (2010). The study was based on

laboratory testing of rapid drawdown in a column prepared with two soil layers; clayey

sand over medium sand. Pore-pressure development was logged by time. The soil

properties considered were—besides permeability—specific gravity, liquid limit,

plasticity index, dry density, void ratio, porosity, and water content at saturation. Some

of the results obtained are shown in Figure 2-3; pore pressures plotted against the

depth, with different curves for different times elapsed. The results did clearly confirm

potential outcomes of rapid drawdown; pore pressures remained high and increased

gradients occurred. Consequences of rapid drawdown, including the fact that pore

Figure 2-3: Results from column rapid drawdown tests performed through a soil sample consisting of a

finer soil layer (clayey sand) put over a coarser soil layer (medium sand). Pore-water pressures at different depths (elevations) are plotted at different times. The soil interface is represented by the dotted line. (Yang et al., 2010)


11

pressures are potentially delayed compared to the external water level, have been

shown to be outward seepage, tension cracks developed, and slope failure (Jia et al.,

2009).

Also rising of a water level might cause problems. In studies concerning the Three

Gorges Dam site (e.g. Cojean & Caï, 2011), reduced slope stability has been noticed

during periods of water-level rise. Generally, water-level rise has been shown to cause

stress redistributions due to external loading (as well increases as reductions, in various

directions), wetting induced issues such as loss of negative pore pressures, and seepage

effects. These changes have been shown to cause loss of shear strength, soil structure

collapse, development of settlements, and slope failure. In a large-scale slope model test

reported, with the external water level being varied, vertical settlements at the crest

were noticed (Jia et al., 2009). The deformation development was almost linear during

the first period of water-level rising, and then continuously declining. The vertical

settlements were assigned to wetting-induced soil collapse. Suction values did abruptly

drop immediately when the water level reached the measure points. The horizontal

total stresses were gradually increasing during the rise. In contrast, the vertical stresses

were gradually reduced; asymptotically leveling out until the end of the rise. The

authors suggested that the reduction of vertical stresses were caused by dissipation of

entrapped air, together with gradually reduced stress concentrations (which were

caused by the early observed differential settlements). The settlements were in turn

explained by the wetting; i.e. entrapped air being replaced by water. Loss of negative

pore-pressures and subsequent loss of shear strength did obviously immediately cause

soil structure collapse/failure. Total stress changes were assigned to homogenization of

the slope body; increased horizontal stresses caused by an increased water load, and

vertical stresses stabilized due to stress-concentration dissipation. It was emphasized that

a delayed change of pore pressure inside a slope—relative to the change of the adjacent

external water level—results in significant movements of water within the slope body.

Thus, the seepage forces were found to adversely affect the stability. The seepage-

instability relationship was also confirmed in Tohari, Nishigaki, & Komatsu (2007).

2.3 Consideration

The earliest tools used to take into account different degrees of submergence of a

slope, involved stability charts (Morgenstern, 1963). In the following decades further

investigations were done utilizing limit-equilibrium (LE) methods for expression of

safety factors (Lane & Griffiths, 2000; Huang & Jia, 2009). Later, these kinds of

problems have also been approached using finite-element (FE) tools (e.g. Lane &

Griffiths, 2000; Pinyol et al., 2011).


12

When it comes to cyclic recurring WLF’s, the fundamentals for each phase of rise and

drawdown, respectively, are obviously the same as for these changes separately

occurring. Though, there are important peculiarities linked to the recurrence. Since

one of the key-issues of evaluating the effects of WLF’s on slope stability concerns

description of groundwater motion, hydraulic modelling is highly important.

Furthermore, any movement of the groundwater level is impacting the geotechnical

conditions; the history of stress and strain changes is central. At the same time, soil

deformations are affecting the pore pressure development. In order to describe this

interaction, and for proper consideration of the behavior of unsaturated soils, fully

coupled hydro-mechanical computations are needed (e.g. Galavi, 2010).

In Huang & Jia (2009) it was stressed that fully coupled consolidation calculations

should be further studied. Coupled hydro-mechanical behavior can possibly be

considered in existing FE-codes. In such processes, calculations of deformations and

groundwater flow with time-dependent boundaries have to be simultaneously carried

out. Since as well saturated as partially saturated conditions have to be properly

handled, consolidation has to be modeled also for unsaturated soils. For description of

unsaturated soils, both elastic-plastic soil-skeleton behavior and suction dependency

have to be considered. The latter concerns both degree of saturation and relative

coefficient of permeability (e.g. Fredlund et al., 1994). Recent models considering

influences of soil mechanical properties on the hydraulic behavior, are usually based on

the dependency of soil-water characteristic curves (SWCC’s), on soil volume, soil

density, or volumetric strain (Sheng, 2011). SWCC’s are relating suction and soil

saturation for a specific soil material (examples are shown in section 4.2.4). Sheng

(2011) did underline the importance of taking into account volume changes taking

place along SWCC’s, when coupling hydraulic components with the mechanical

components in constitutive models. The author stated that neglecting this volume

change could mean inconsistent predictions of changes of volume and saturation.

Among studies applying hydro-mechanical coupling, many have been addressing issues

occurring in sediment-transfer systems. These are generally focused on sediment

loading problems, sources of contributes to a river’s sediment budget, changed river

shapes caused by sedimentation etc. (e.g. Darby et al., 2007; Fox et al., 2007; R.

Grove, Croke, & Thompson, 2013). In Darby et al. (2007) a method for performance

of a coupled simulation of fluvial erosion and mass wasting was presented. The method

did consider the dynamics of bank erosion involved coupling a fluvial-erosion model

with FE-seepage analysis and LE-stability methods. The mass wasting was simulated to

occur as a series of failure episodes. The study was limited to cohesive riverbanks, and

mainly aimed to find a method to properly quantify bank derived sediment volumes.


13

Though, the modelling did involve both particle-size characterization and soil-strength

consideration.

The shortfall notwithstanding, some studies on large-scale reservoir fluctuations are

found in the literature. For instance, it was found that the stability of a reservoir slope

consisting of sand and silt, was more directly governed by hydraulic conductivities than

by velocities of water-level changes (Liao et al., 2005). For evaluation of reservoir-

fluctuation effects on a silty slope, the importance of negative pore pressures, friction

resistance, and water-load support was highlighted in Zhan, Zhang, & Chen (2006). In

that study, saturated-unsaturated seepage analysis was combined with LE-stability

analysis. Also in Shen, Zhu, & Yao (2010) examination of “reservoir water-level

fluctuation” did include analysis of only one cycle of rise and drawdown, carried out

by means of LE-calculations. In Galavi (2010) hydro-mechanical theories used in FE-

modelling were presented. Comparisons and evaluations performed showed good

agreement between the results obtained from the numerical FE-computations

performed using the FE-code PLAXIS 2D and those from analytical solutions. In

Kaczmarek & Leśniewska (2011) effects of groundwater-level changes on a flood bank

core were modeled. Stability and seepage were considered by FE-analysis. Though, the

study was only briefly described and neither the input nor the outcomes were

satisfactorily presented.


14

Slope analysis

15

3 SLOPE ANALYSIS

3.1 Introduction

Historically, methods for evaluation or determination of slope stability have been

largely focused on theories regarding fine-grained soils. During the 1840s—related to

the railway construction projects ran at that time—engineers were working on

determination of the shear strength of clay (Ward, 1945). The analysis approach of

assuming circular slip surfaces was introduced in 1916 (Petterson, 1955). Though, a

fully described method was presented by Fellenius in the beginning of the 1920s (e.g.

Krahn, 2003; J. M. Duncan & Wright, 2005). In 1945, Skempton was presenting a

study named “A slip in the west bank of Eau Brink Cut”, including an illustration of a

rotational slip surface within a slope mainly consisting of clay (Ward, 1945).

During the past decades, engineering and research studies have brought knowledge and

formulated theories within the area of soil mechanics. The understanding of different

factors affecting slope stability (e.g. time dependent changes of soil behavior), has been

continuously improved. This not at least due to known limitations of existing methods

for evaluation of soil strength (e.g. laboratory and in situ testing methods) and

development of new tools and instruments for measuring and observing slopes. All this

together has successively formed the now available collection of experience and

knowledge concerning slope behavior and slope failure. This applies for improved

understanding of basic principles of soil mechanics, and improved analysis procedures.

(J. M. Duncan & Wright, 2005)

3.2 Description and classification of soil

The soil skeleton consists of three components. The solids consist of mineral grains

and/or organic constituents. Moreover, free spaces between the solid grains—i.e. the

pores/voids—are exhibiting properties (size and shape) depending on the geometrical

properties of the solids. The pores are filled with fluids (often pore water), gas (often

air), or a mixture of these two. The geotechnical properties of a soil volume are

depending on minerals being present, content of organic material, content of water,

manner of which the water is present, denseness of the soil particle arrangement, and

size and shape of the particles. Different soil types are then grouped with respect to

these characteristics.


16

Soils are usually divided into two main groups; fine-grained soil and coarse-grained soil.

This distinction is based upon the particle sizes; the diameters, d. The fractions clay and

silt are sorted as fine-grained soils, whereas sand and gravel are sorted as coarse-grained

soils. Soils are also classified based upon the kind of forces being contributing to the

soil strength. The grains in purely cohesive soils are held together by van der Walls

forces, hydrogen bonds, and electro-static forces; the resultant makes up cohesive

bonding. On the other hand, the strength of sand and gravel, i.e. frictional soils, is

coming from mechanical inter-granular friction. Silt is sometimes with respect to

strength properties called intermediate soils. (Craig, 2004)

3.3 Stress and strength

3.3.1 Stress

Classical theory

In contrast to fine-grained soils (with strength coming from cohesive forces), the

strength of a coarse grained soil mostly comes from contact forces between the

individual grains. The mass of the soil particles, together with any additional forces

from external loads, are getting normal forces acting at the particle contact areas. The

resultant forces acting on a soil element are to be transmitted through the soil skeleton

components, including also fluids and/or gases. The fundamental and well established

principle of effective stress, , presented by Terzhagi in the 1920th, is expressed as:

where is the total normal stress, and is the pore-water pressure. This equation is

based upon the assumption that there are only two situations; (1) there is a pore

pressure coming from the fully water filled pores, and (2) there is no pore pressure.

Unsaturated soil behavior

In the middle of the 1950th, Bishop proposed an effective stress expression, considering

also partially saturated or unsaturated soils:

( ) (3.2)

where is the pore-air pressure, the pore-water pressure, and an

experimentally determined matric-suction coefficient. This coefficient (Bishop’s

parameter) could be related to the degree of saturation, . The experimental

evidences of are sparse (Galavi, 2010), and the role of the parameter has been

(3.1)

Slope analysis

17

generally questioned (Fredlund, 2006). The often used assumption of letting the

matric-suction coefficient be replaced by the degree of saturation, i.e. , has been

shown to require care to be taken. The validity of the assumption is highly dependent

on the microstructure of the soil type being analyzed; more specifically the fraction of

water filling the micropores, (Pereira et al., 2010). The deviation has been shown

to be higher for fine-grained soils, especially those exhibiting high plasticity. Though,

in PLAXIS (2012) the matric-suction coefficient is substituted with the effective

degree of saturation, . Assuming air-pore pressure is being negligibly low, the

effective stress expression (equation (3.2) becomes:

(3.3)

Unsaturated soil behavior is further discussed in section 4.2.

A number of proposals have been presented aiming to improve the accuracy regarding

definition and description of stress-strain behavior and strength of unsaturated soils.

Limitations of existing effective stress expression were indicated by e.g. Burland &

Jennings (1962). The possibility to control the mechanical response of unsaturated soils

by considering two stress-state variables, namely (1) the net stress, i.e. the difference

between the total stress and the pore-air pressure, and (2) the suction, i.e. the

difference between the pore-air pressure and the pore-water pressure, was firstly

demonstrated by Fredlund & Morgenstern (1977). Moreover, a number of non-linear

elastic constitutive models were developed based upon this. Subsequently, also elastic-

plastic models have been proposed and developed (Sun et al., 2010). Many models are

developed based on the Basic Barcelona Model (Alonso, Gens, & Josa, 1990). The subject

of unsaturated soil was well reviewed in Sheng (2011).

3.3.2 Particle arrangement

At a loose soil-particle arrangement, only a small lateral force is needed to get the soil

particles rearranged into a denser configuration. Therefore, loose soils exhibit low

resistance to volume decrease (contractancy) and deformations. Horizontal shearing of a

dense coarse grained soil sample means stiff and almost elastic initial response. This

since the soil particles of a dense soil are forced to “climb” over each other (in the

direction against the applied vertical load), in order to get a rearrangement taking

place. The soil undergoes volume increase (dilatancy). The response of shear stress, at

increased shear straining, is conceptually shown in Figure 3-1. When shearing

takes place without any further volume change, the critical state is reached.


18

Figure 3-1: Shear stress/shear strain development for soils of different denseness. (after Hansbo, 1975)

3.3.3 Strength and failure

Since the shear stress is directly dependent on the present effective normal stress, the

shear strength is directly dependent on the effective normal stress at failure. One of the

most known and used hypotheses for description of shear failure is Mohr-Coulomb´s

theory, expressed by use of the strength parameters friction angle, and cohesion,

(or and for drained conditions). Graphically, the failure criterion might be

presented in the Mohr-Coulomb plane; either drawn as a straight line being a tangent

to the shear stress/normal stress envelope (tangent parameters), or drawn from the origin

to one specific stress point (secant parameters), see Figure 3-2. In the former case the

strength parameters are valid only for a stress range around the touching point, whereas

the ´-value in the latter case is only valid for one single stress state. According to the

normally used tangent approach, the shear strength, is expressed:

(3.4)

where c´ and ´ are valid for a limited effective stress range. The cohesion intercept, ’

obtained from the evaluation described is not to be confused neither with the “real”

cohesion—i.e. the strength contribution originating from the inter-particle cohesive

forces—nor the apparent cohesion, arising either from capillary tension stresses

(negative pore pressure) or inter-granular cementation.

Slope analysis

19

Figure 3-2: The bold solid curve connecting the plotted stress points shows the actual shear stress-normal stress combinations of a specific soil. The thin solid line represents the envelope used for tangent approach evaluation of the strength parameters, whereas the dashed line represents the secant approach. (after Craig, 2004)

3.3.4 Angle of internal friction

For a drained granular soil, the total angle of friction—i.e. the angle of friction

mobilized during shearing, —is actually consisting of two main parts:

(3.5)

where is the inter-particle sliding component and the geometrical interference

component. The geometrical component is, in turn, consisting of two parts, namely

coming from the dilation phenomenon (particle climbing), and coming from

particle pushing and rearrangement (Terzaghi et al., 1996).

The final expression is moreover:

(3.6)

3.3.5 Factors governing internal friction – changed properties

Mineral composition and crushing properties

The soil-particle minerals are affecting a coarse-grained soil by two factors; (1) the

friction of the particle surface is directly dependent on the surface roughness (affecting

the sliding component ), (2) the mineral hardness and strength is affecting the

crushing properties of the individual particles (affecting the component ) (Terzaghi

et al., 1996).


20

Dilatancy

The dilatancy angle, is used to describe volume increase during shearing. At low

confining stresses, particles are moving as well through particle pushing as through

particle climbing. This means an increased geometrical component (up to 30°)

Terzaghi et al. (1996). According to Rowe´s theory (Rowe, 1962), the dilatancy

phenomenon could be expressed:

(

)

(

) (3.7)

where and are the major and the minor principal stress, respectively, and

is the rate of dilatancy (stresses and strains are further described in section

4.1). The index cv indicates critical state. The approach was based on results from

compressional triaxial tests and the theory of Mohr circles. There are many studies that

have been aimed to properly correlate the dilatancy angle to the friction angle. In

Bolton (1986) it was stated that simple expressions are preferably used. One such

simplification is the relationship (where is the friction angle at

critical state). A simplified model is shown in Figure 3-3.

Figure 3-3: A soil volume subjected to a normal force, N and a horizontal shear force, T undergoes

dilatancy. The dilatancy angle is denoted , the friction angle , and the friction angle at critical state

. (after Axelsson, 1998)

Even though this principle way of illustrating dilatancy is based on a number of

assumptions, e.g. the particles being perfectly spherical (and of the same size), the

fundamental mechanical effect is well described. Though, according to the definition

of the critical state—i.e. that the dilatancy is zero—Bolton (1986) stated that the simple

way of describing the dilatancy as above, means an overestimation of the difference

by approximately 20%, compared to the Rowe´s stress-dilatancy relationship,

and suggested that the relation:

Slope analysis

21

(3.8)

fits Rowe´s stress-dilatancy relationship well. In PLAXIS (2012) the definition of

dilatancy angle is based on Rowe´s theory:

( )

(3.9)

where is the mobilized friction angle and is the friction angle at critial state.

Void ratio and grain size distribution

Depending on to what extent the particles are interlocked and clamped by each other,

the shearing resistance will vary. A simple measure of the soil-skeleton structure is the

void ratio, , defined as the ratio of pore volume, and the volume of the solids, .

At small void ratios of a coarse-grained soil, the total contact area between the particles

is large; the friction angle is high. Therefore, in well-graded soils (e.g. glacial-tills),

where smaller grains are filling the larger voids, friction angles are generally higher than

in poorly graded soils (e.g. uniform sands).

Particle shape

Since Wadell (1932)—one of the first to put attention on how particle shape is

affecting soil behavior—a number of studies have been carried out. In Krumbein

(1941) as well the importance of considering the particle shape, as some methods for

determination and analysis, were discussed. Connections and correlations between

particle shape and mechanical behavior of soils—strength, stiffness/deformations, and

hydraulic properties—are described and discussed in soil mechanics literature (e.g.

Terzaghi et al., 1996; Guimaraes, 2002; Santamarina & Cho, 2004; Mitchell & Soga,

2005; Cho, Dodds, & Santamarina, 2006).

Although there seems to be an agreement regarding the existing influence, the particle

shape is rarely considered in designing of geotechnical structures. Even though particle

shape—as an influencing factor—is mentioned in some guidelines and designing codes,

e.g. Eurocode 7 and RIDAS (Swedenergy, 2008), it is not clearly described how to

actually apply particle-shape information. (Johansson & Vall, 2011)

In literature, the term shape is found to be used some inconsequently and there are

significant variations in how different sub-quantities are used (Johansson & Vall, 2011).

One common approach used for handling the complexity of particle-shape description,


22

is based on three sub-quantities of different geometric scales (Wadell, 1932; Krumbein,

1941; Mitchell & Soga, 2005). At large scale—i.e. when the particle diameters in

different directions are described—the term sphericity is used. At an intermediate

scale—for description of to what extent the particle boundary/perimeter is deviating

from a smooth circle or ellipse—the sub-quantity roundness is used. To describe the

smallest scale of the particle—the texture of the particle surface—roughness is used.

For description of connections between soil-particle shape and the influenced

properties, there are a number of empirical relationships developed. An often seen

expression relating particle shape with strength is:

(3.10)

where v

is the friction angle at critical state, and is a quantity describing the

roundness of the particles. This relation shows that the critical state friction angle is

decreasing with increased roundness of the soil particles (Santamarina & Cho, 2004;

Cho et al., 2006). Also void ratio is depending on the particle shape. At large strains,

when the particle displacements are coming from particle rotation and inter-particle

sliding, the void ratio is decreasing with increased roundness and increased sphericity

(Cho et al., 2006). Since the strength properties are linked to void ratio (and/or

porosity), also this relation affects the strength of the soil. The relevance of

relationships between particle shape and mechanical soil properties, including the

possibility to quantify the shape, was discussed in Rodriguez, Edeskär, & Knutsson

(2013).

3.4 Slope-stability analysis

3.4.1 Introduction

Among the factors causing reduced strength of a soils mass, increased pore pressure is very

important. In addition to pore-pressure increase, also seepage—i.e. water flowing

within the soil—might lead to washing out of soil particles, which moreover cause

undermining of the soil (Ward, 1945). A lot attention has been put on natural events

entailing an upward movement of the groundwater level; e.g. precipitation and

subsequent infiltration (e.g. Lim et al., 1996; Tohari et al., 2007). Another

phenomenon potentially resulting in strength loss is degradation and/or weathering;

physical, chemical, and biological processes breaking down soil particles, rearranging

the structures, and changing the properties of the entire soil mass (J. M. Duncan,

2005).

Slope analysis

23

The slope stability may also get reduced by increased shear stress due to changed loading

conditions. The shear stress could be increased by (1) increased stresses acting in

directions being driving; external loads at the top of the slopes, water pressure in cracks

at the top of the slope, increased soil weight due to increased water content etc. The

shear stress could also be increased by (2) factors making the resistant forces reduced;

any kind of removal of soil at the bottom of the slope, and drop in water level at the

slope base. (J. M. Duncan, 2005)

As mentioned in previous sections, there are different approaches available for

performance of stability analyses; limit-equilibrium methods, and deformation-analysis

methods.

3.4.2 Limit-equilibrium analysis

Limit-equilibrium (LE) methods have been used to analyze slope stability for a long

time (J. M. Duncan & Wright, 1980; Yu et al., 1998; Zheng et al., 2009). These

methods have been unchanged for decades (Lane & Griffiths, 1999). In J. M. Duncan

& Wright (1980) it was emphasized that all equilibrium methods for slope-stability

analysis have some characteristics in common: (1) The factor of safety (FOS) is defined

as the ratio between the shear strength, and the shear stress required for equilibrium,

. Using Mohr-Coulomb’s failure criterion and also applying Terzaghi´s expression for

effective stress, the factor of safety becomes:

( ) ( ) ⁄ (3.11)

where normal stress and pore pressure are denoted and u , respectively, and the

strength parameters friction angle, and cohesion (expressed in terms of effective

stresses, i.e. for drained situations) are denoted by , and , respectively. (2) It is to be

assumed that the stress-strain characteristics of the soils are non-brittle, and that the

same shear-strength value may be mobilized over a wide range of strains along the slip

surface. (3) Some (or all) of the equilibrium equations are used to determine the

average shear stress, and the slip surface normal stress; this in order to possibly use

Mohr-Coulomb’s failure criterion for shear strength calculation. (4) Since the number

of unknowns is larger than the number of equilibrium equations, explicit assumptions

are to be involved.

Among the single free-body procedures, i.e. the group of the LE-method that was first

used, there are in turn sub-procedures being based on different approaches; e.g. the

infinite slope procedure, the logarithmic spiral, and the Swedish circle method. For all of the

single-free body procedures, homogenous soil profiles are preferred. External water


24

levels are considered as external loads, and the groundwater level determines from

what level the soil will get saturated conditions. In Figure 3-4 the principles of some

single free-body limit equilibrium procedures, are shown. In the infinite plane case, the

slope inclination, the soil weight, and the strength parameters are needed for

expression of the factor of safety. For the two other procedures, the shape of the

assumed slip surface, the mass of the soil block bounded by the slip surface and the

ground surface, and the location of the center, are to be determined. (J. M. Duncan &

Wright (2005)

(a) Infinite slope procedure (b) Logarithmic spiral

procedure

(c) Swedish circle procedure

Figure 3-4: Principle sketches of the basics of three single free-body limit-equilibrium procedures. The

ground surfaces, GS and slip surfaces, SS are shown. Moreover, also slope inclination for the infinite

slope case, ; the initial radius, for the logarithmic spiral case; and the constant radius, for the Swedish circle case, are presented.

Slice procedures

Slices procedures were introduced already in the end of the 19th century (Ward, 1945). In

order to simplify the process of determining the driving moments, and to make it

possible to consider slip surfaces of more complex geometries, the soil block could be

divided into vertical slices. Forces acting on a typical slice are weight, ; shear force

on the slice base, (defined as , where is the shear strength and is the

length of the slice base); effective normal force acting on the base, ; water pressure

force acting on the base, ; vertical side force, ; horizontal side force, (see Figure

3-5).

In the simplest slice procedure, often referred to as the ordinary method of slices, only

overall moment equilibrium is considered. In Bishop´s simplified method also the

equation of vertical force equilibrium of each slice is satisfied; in Bishop´s modified

method, also side forces are considered. All these methods are restricted to circular slip

surfaces. Moreover, there are methods where only the force equilibrium equations are

satisfied; e.g. a method presented by Lowe and Karafiath in 1960 (J. M. Duncan &

Slope analysis

25

Wright, 1980). Methods satisfying as well moment equilibrium as force equilibrium

(e.g. Morgenster a d Pri e’s method, Spe er’s method, and Ja bu’s ge eralized pro edure o

slices) are applicable on slopes with slip surfaces of any geometry. When stability

analyses are performed using LE-methods, consideration of pore-pressure situations are

often considered by simplified methods (Huang & Jia, 2009). This might be done by

assuming hydrostatic pore pressures (if there is no flow), use of flow-nets (for steady-

state seepage), approximate charts (for transient/unsteady-state seepage), or pore-

pressure ratios (J. M. Duncan, 1980)(as an alternative tool for steady-state seepage

consideration or for analytical consideration of development of excess pore pressures

due to consolidation).

Figure 3-5: A typical slice with acting forces. (J. M. Duncan & Wright, 1980)

Still, it is worth mentioning that drawbacks and limitations of LE-methods have been

known and evidenced for a long time. For instance, there are difficulties connected to

accurate definition of the position of the slip surface (Ward, 1945), and there are

numerical problems potentially met with Bishop´s modified method (J. M. Duncan,

1996). An illustrative example of what kind of reasoning being characteristically used

to justify utilization of LE-analysis was seen in Tsai & Yang (2006); an LE-based study

focused on landslide prediction combined with consideration of rain-triggering effects.

Several authors were referred to when it was stated: “the infnite slope stability analysis

is a preferred tool to evaluate landslides due to its simplicity and practicability”.

3.4.3 Deformation analysis

As a consequence of known drawbacks of LE-procedures, together with a

computational capacity being continuously improved, more accurate methods have

been gradually more requested and developed. Numerical modelling approaches have


26

been most widely used based on finite-element (FE) methods; applied to the field of

soil mechanics in the 1960s (J. M. Duncan, 1996). When using FE-methods the

volume/area being analyzed is considered as a continuum whereupon it is discretized

to a finite number of sub-volumes/sub-areas (elements) forming a mesh. In contrast to

the fundamentals of LE-methods, a system analyzed with deformations-analysis

approaches is not considered to necessarily be in equilibrium; neither in terms of forces

nor moments. Besides consideration of stresses and geometrical conditions, stress-strain

relationships are to be described. This is done by use of constitutive models considering

different material properties. Moreover, the key-issue is to as reliably as possible define

these constitutive relationships for a certain soil.

When it comes to definition of stability in terms of safety factors, a procedure of

simulating reduced strength until failure occurs is used. This method was firstly used in

the 1970s (Zheng et al., 2009) and is usually referred to as the shear strength reduction

technique. In PLAXIS 2D 2012 this is mathematically handled by expressing ratios of

the strength parameters:

(3.12)

where is a controlling multiplier representing the safety factor; is the cohesion;

is the friction angle; and the index reduced indicates values being successively

decreased until failure occurs. For indication of global instability of soil slopes, the non-

convergence of solutions is often used. This is indicated by the condition when a stress state at which the failure criterion and the global equilibrium can be simultaneously

satisfied, is not found (Lane & Griffiths, 2000).

The method is usable for constitutive models utilizing strength parameters that can be

reduced. The safety factor is principally defined as in LE-approaches, i.e. by the ratio

between the available shear strength and the shear stress at failure.

3.4.4 Drainage conditions

Since soil strength is directly dependent on the presence of water and the interaction

between the soil and the water, consideration of draining conditions is crucial for proper

evaluation of different soil-mechanical behavior patterns. In a situation with fixed

conditions—i.e. concerning water level, loading, inherent soil properties, and

geometry—the pore pressure is hydrostatic. At occurrence of any changes, the pore-

pressure situation is potentially affected. In coarse-grained soils, water is easily moving.

Slope analysis

27

This means that loading could be allowed without the pore-pressure state being

changed; drained analysis is performed. In fine-grained soils, low permeability might

mean a rate of drainage being lower than the rate of total stress change, meaning

potential development of increased (excess, non-hydrostatic) pore pressures.

Analogously, this could also be the case in coarse-grained soil if the total-stress is

changed rapidly enough. In the two latter situations, undrained analysis is to be

performed. Depending on what kinds of changes being expected and for what time-

perspective the stability is assessed, either drained or undrained conditions are most

critical. Therefore, both cases are to be considered; combined analysis is performed.


28

Modelling of soil and water

29

4 MODELLING OF SOIL AND WATER

4.1 Constitutive description of soil

4.1.1 Stress

For each specific stress state there are orthogonal stress directions in which the normal

stresses display their maximum and minimum values; principal stresses, acting in the

principal stresses directions. At the planes directed perpendicular to the principal stress

directions, no shear stresses are acting. In two-dimensional (plane stress) situations there

are two principal stresses; the major principal stress, and the minor principal stress,

. In a three-dimensional situation the minor stress is denoted , and the

intermediate principal stress gets the denotation .

The average value of the principal stresses acting in a specific point within the soil mass

is governing the volume change of a soil unit. It is termed mean pressure, p´, expressed

as:

( ) (4.1)

The stress measure deviatoric stress is governing the shape change of a soil volume, for

the triaxial case expressed :

√

(

) (4.2)

where , and are the deviatoric normal stresses components; each defined as the

difference between the mean pressure and the corresponding principal stress.

4.1.2 Strain

Normal strain, which indicates the displacement in a specific direction, e.g. in the x-

direction, is—according to engineering notations—defined as:


30

(4.3)

where is the strain, is the displacement, and is the initial length. Strains in

other directions are analogously expressed. The shear strains are moreover (for small

strains)—again according to engineering notation—determined by summing the ratios

of displacements and initial dimensions perpendicular to the measured displacements,

e.g.:

⁄ ⁄ (4.4)

where is the shear strain component in the x:y-plane, is the displacement

in the direction perpendicular to the length/dimension , ( and analogously

related).

In order to capture volume changes governed by isotropic compression, the volumetric

strain increment, is expressed as:

(4.5)

where the ratio of a negative change of the specific volume, and the initial

specific volume, , is defined as a positive volumetric strain. Specific volume is

defined as (where is the void ratio). The total volumetric strain is

expressed as a sum of the principal strains ( ) or as a ratio of the

volume change and the total volume ( ⁄ ).

Deviatoric normal strains are associated with shape changes. In the x-direction, is

expressed as:

(4.6)

with the variables defined above. Deviatoric normal strains in other directions are

analogously expressed. In case of triaxial loading—with an axial strain, deviating

from the two radial strains , the deviatoric axial strain, (also denoted ) becomes:

( )

(4.7)

4.1.3 Elastic response

Stress-strain relationships for a soil are different for compression (volume change) and

shearing (shape change). At axial or isotropic compression, axial (index: a) and/or


31

volumetric strains (index: vol) are developed; in both cases the response is non-linear

(Figure 4-1). The fundamental shear-response curves from as well direct shear testing

(axis labels in brackets) as triaxial testing at drained conditions are presented in Figure

4-2.

Figure 4-1: To the left the non-linearity is sown (curves for one-dimensional or isotropic compression); to the right the dissimilarity of the response of fine-grained and coarse-grained soils is shown (compression curves plotted in a logarithmic-linear diagram). (after Axelsson, 1998)

Figure 4-2: Response curves from shearing of soil at loose arrangement (curve 1), dense arrangement (curve 2), and critical arrangement (curve 3). Quantities for direct shearing are presented within brackets

whereas the triaxial test notation is presented without. The strains are indexed e (elastic) and p (plastic). (after Axelsson, 1998)

It is seen that the maximum deviatoric stress (or shear stress), mobilized during

shearing, varies depending on the soil grain arrangement prior to loading. For a soil

with dense grain arrangement—i.e. a soil that undergoes dilation during shearing—the

maximum deviatoric stress is larger than the critical one. This maximum is reached at

strains smaller than for as well loose as critically configured soils.


32

The resistance against elastic straining is described by stiffness moduli. The description

used in the Mohr-Coulomb’s elastic-perfectly plastic model, is presented in section

4.1.6.

4.1.4 Unloading/reloading

Once a soil is unloaded, it will undergo swelling (corresponding to elastic

deformations). The subsequent reloading of the same soil will result in a stress-strain

response that coincides with the unloading path. To curve 1 in Figure 4-2 an

unloading/reloading-curve (dashed), and a straight line that approximately represents

the unloading/reloading-curve, is added.

4.1.5 Plastic response

For stresses high enough, compression or shearing of a soil ends up with irreversible

(plastic) strains. In such a situation, the total strains developed due to loading consist of

as well elastic as plastic strain contributions (see Figure 4-2). The response is therefore

said to be elastic-plastic. In the x-direction this is expressed as:

(4.8)

where the total strain increment, consists of the elastic increment, and the

plastic increment,

. The strain increments in other directions are analogously

expressed.

The soil behavior related to elastic-plastic deformation is captured by the theory of

plasticity. In order to properly take into account both elastic and plastic strains when

analyzing or modelling soil behavior, the stress states at which plastic strains are

developed, have to be defined. This limit—the yield criterion—is expressed by a yield

function, f defining the stress limits in one, two, or multiple dimensions. For instance,

regarding one-dimensional (or isotropic) compression, the yield function is depending

on the effective mean pressure, ; more specifically the preconsolidation pressure, .

For stresses higher than the preconsolidation pressure, there will be plastic strains

developed. For multiaxial stress states, the yield functions can be expressed as e.g.

( ) or ( ) Geometrically, the yield criteria in the stress plane

and the stress space are represented by a yield locus, and a yield surface, respectively. The

yield criteria are expressed in different ways in different elastic-plastic constitutive

models. Though, stress states entailing yield function values mean development

of only elastic deformations. For these states the strains are determined using elastic

stiffness moduli, in addition to values of mean stress and deviatoric stress.


33

At stress states causing the yield criteria being reached ( )—i.e. when plastic strains

are developed—the strain magnitude is to be determined using a flow rule based on the

gradient of the so called plastic potential function, . Moreover, the plastic strain increment,

is expressed as a function of the partial differential of the plastic potential function,

(differentiated with respect to the stress). Depending on the type of deformation

being of interest, the strain increments are expressed in different ways. In Table 4-1

common strain-increment expressions are presented; including the plastic potential

functions related to each expression.

Table 4-1: Strain-increment expressions and plastic potential functions for different stress quantities. The plastic potential functions contain at least the variables presented in the table.

Stress quantity Strain

increment 1) Plastic potential function, 2)

Normal stress

(engineering notation)

( )

3)

Normal stress

(tensor notation)

( )

4)

Mean pressure

( )

Deviatoric stress

( )

1) The plastic multiplier, determines the magnitude of the plastic straining. 2) The plastic potential functions are depending on at least the variables presented in the table.

Different constitutive models may involve addition variables. 3) Increments in other directions are analogously expressed by substitution of x. 4) General tensor notation. The partial differential of the plastic potential function is determining

the direction of plastic straining.

Since as well the yield function as the plastic potential function are functions of the

stress state, there might be cases when . This special case is termed associated flow.

This kind of flow is used when contractancy response is significant (e.g. in normally

consolidated clays). Still, the general case—when , i.e. non-associated flow—is

frequently used to describe the behavior of coarse-grained soil where both

contractancy and dilatancy are possibly occurring.

For elastic-plastic models there are different approaches for definition of the yield

function. The classical way of describing elastic-plasticity is based on so called elastic-

perfectly plastic theory. In these approaches the yield functions are assumed to be fixed,

i.e. not influenced by the plastic straining itself. In contrast to this, there are also more

sophisticated models which are considering yield property changes related to


34

development of plastic strains. As a consequence of strain dependency of the yield

function, the stress limit at which plastic strains are initiated, changes. This means that

the elastic range will either grow or shrink as a result of developed plastic strains. For as

well one-dimensional as isotropic compression, hardening (grown elastic range) is

simply described by an increased preconsolidation pressure, . This means that the

yield function is also dependent on this quantity, i.e. ( ).

Time dependent strain

In order to fully describe plastic yielding, time related phenomena are to be

considered. This kind of yielding—i.e. permanent deformations developed over time,

under constant effective stresses—is usually referred to as creep. In Tatsuoka (2007) it

was emphasized that an increment of plastic strain, is caused by three phenomena;

(1) plastic yielding, (2) viscous deformation (i.e. time-dependent yielding), and (3)

inviscid cyclic loading effect. usually called creep. Historically, the viscous deformation

part (creep) has mostly been considered as an issue in fine-grained soils; especially in

highly plastic clays. For fine-grained soils creep is mainly caused by volumetric

yielding; usually explained by “secondary consolidation”. However, when it comes to

coarse-grained soils—i.e. where the deformations are mainly caused by shear

yielding—the research has been relatively limited. The lack of studies performed on

sand has been noticed and expressed by many authors, e.g. Murayama, Michihiro, &

Sakagami (1984), Lade & Liu (1998) and Kuwano & Jardine (2002).

Though, in the late 70s and early 80s, some researchers were carrying out experimental

studies on creep characteristics of sand. Rheological models were formulated and

evaluated, and it was suggested that creep behavior of sands could be expressed in

terms of stresses and stress ratios; the findings were presented in e.g. Murayama et al.

(1984). During recent years there have been some more studies carried out on time-

dependent deformation on coarse-grained soils. Kuhn & Mitchell (1993) did propose a

distinction to be made between two main groups of mechanisms to be considered

when it comes to creep rate changing with time; deformation-dependent mechanisms and

intrinsically time-dependent processes. Di Benedetto, Tatsuoka, & Ishihara (2002) presented

how to consider also viscosity (i.e. creep behavior) in an elastic-plastic model used for

description of the behavior of sand. As a fundamental part of the model presented, a

distinction was made not only between elastic and plastic strains, but also between

time-independent and time-dependent stresses. The instantaneous stress, was

expressed as:

(4.9)


35

where the inviscid stress component, is time-independent but still a function of the

instantaneous irreversible (plastic) strain, and the viscous stress component, is a

function of as well and its rate (i.e. ). Moreover, the “time-dependency”

of is connected to the history of , rather than to the time, . The authors were

stating—also based on earlier conclusions (Tatsuoka, 2000)—that the time, is not a

relevant parameter for geomaterials. Since the plastic and the viscous strains are not

independent of each other, Tatsuoka (2007) suggested a visco-plastic strain

component being a part of a non-linear three-component rheology model (Figure

4-3). In Peng et al. (2012) a study on FEM-modelling of a dense sand loaded by a strip

footing was presented. The study was mainly aimed to capture viscous behavior before

peak strength being reached. The results were found to be in agreement with physical-

test data. Though, the testing-time was in the order of hours.

Figure 4-3: A non-linear three-component rheology model considering the dependency between viscous

and plastic strain: elastic strain rate denoted , and visco-plastic strain rate denoted . (after Tatsuoka, 2007)

4.1.6 Mohr-Coulomb model

This section is focused on the model descriptions in PLAXIS (2012). This model was

used for description of the soil behavior in the modelling work done in this study

(Paper B).

The Mohr-Coulomb model is an elastic-perfectly plastic constitutive soil model. The

model is based on Coulomb´s friction law and describes the stress-strain relation either

as fully elastic or fully (perfectly) plastic. The elastic behavior is described by Hook´s

law i.e. by use of the parameters Young´s modulus, and Poisson´s ratio, .

The Young´s modulus is generally expressing the relation between stress and strain, i.e.

the stiffness. Due to the non-linearity of stress-strain response of soils, the stiffness is to

be defined in different ways depending on the specific stress-strain state being analyzed.

For standard drained triaxial tests, the initial stiffness is usually denoted , defined as a

tangent to the shear curve. Due to the very small range of linear elasticity of soils, this


36

modulus is not properly describing the stiffness of loaded soils. Therefore, the modulus

is instead used. This is a secant modulus intersecting the shear curve at the

deviatoric stress corresponding to 50% of the peak value. Moreover, elastic

unloading/reloading is described by the modulus . In Figure 4-4, the definitions of

the moduli are principally shown.

Figure 4-4: The initial stiffness , the often used secant stiffness, , and the unloading/reloading

stiffness, , defined. The unloading/reloading curve is represented with the long-dashed line. (modified after PLAXIS, 2012)

Poisson´s ratio, expresses the ratio of strains developed perpendicularly to each other.

For triaxial compression, the parameter expresses the ratio between lateral and axial

strains. For one-dimensional compression, the relationship between Poisson´s ratio,

and the horizontal and vertical effective stresses, is expressed as:

(4.10)

The stiffness can alternatively be described and defined by the shear modulus,

defined as:

( ) (4.11)

or by the oedometer modulus, defined as:

( )

( )( ) (4.12)


37

Since the failure line defines the ultimate state up to which the developed strains are

elastic, this fixed line is also representing the yield function, separating the two types of

strain response (elastic and plastic). The full three-dimensional yield criterion consists

of six yield functions; two functions in each plane of stress pairs. In terms of principal

stresses, and with the major and the minor principal stresses ( and ), the yield

functions are expressed as:

( )

( ) ( ) ( ) (4.13)

and

( )

( ) ( ) ( ) (4.14)

Moreover, there are another four equations needed (analogously expressed, with all

combinations of and ) to fully define the yield surface in three dimensions.

The surface has the shape of a hexagonal cone, illustrated in Figure 4-5.

In the plastic potential functions, which are parts of the flow rule (see Table 4-1) an

additional plasticity parameter is used; the dilatancy angle, (discussed in section

3.3.5). The plastic potential functions are expressed as:

( )

( ) ( ) (4.15)

and

( )

( ) ( ) (4.16)

Moreover, there are another four equations needed (analogously expressed, with all

combinations of and ) for definition the whole yield surface.


38

Figure 4-5: Mohr-Coulomb´s yield surface. (PLAXIS, 2012)

4.2 Hydraulic modelling

4.2.1 Fundamentals

The degree of which a soil volume consist of pores is defined as porosity, , expressed

as:

⁄ (4.17)

where is the pore volume and is the total soil volume. The degree of which the

total pore volume in a soil element is filled with water is defined as the degree of

saturation, and according to classical soil mechanics expressed as:

⁄ (4.18)

where is the water volume and is the pore volume.

Moreover, the soil unit weights are defined according to the input values of

unsaturated weight, for the soil located above the phreatic level, and saturated

weight, for the soil below.

4.2.2 Unsaturated soil

When considering unsaturated soil, the effective degree of saturation, is defined as:

( ) ( ) (4.19)


39

where is the degree of saturation and the indices max and min denote maximum and

minimum values. The effective degree of saturation is used also for definition of actual

soil unit weight, :

( )

(4.20)

where the indices denote unsaturated and saturated conditions.

4.2.3 Flow

The rate of water flow, in saturated soil is according to Darcy´s law defined as:

(4.21)

where is the hydraulic gradient (defined as the ratio between head loss, and the

flow length ), ( ⁄ ), is the water density, is the acceleration of

gravity, is the intrinsic permeability, and is the dynamic viscosity of the fluid (the

water).

In unsaturated soils, the hydraulic conductivity is reduced due to air present in the

pores. This reduction is in PLAXIS 2D 2012 (Galavi, 2010) considered by expressing

the specific discharge, , as:

( ) (4.22)

where is the gradient of the pore pressure causing the flow; is the vector of

acceleration of gravity; and is the relative permeability, defined as:

⁄ (4.23)

where is the tensor of permeability at a given saturation. Moreover, is in van

Genuchten model computed according to the dependency on , see equation (4.27).

Starting with the continuity equation—involving mass concentration ( ) and the

divergence of the mass flux density of the residual water ( )—neglecting the

compression of the solid particles as well as of the overall soil structure, the expression

for transient flow becomes:


40

(

)

[

( )] (4.24)

where is the bulk modulus of water, and other parameters as above. In case of

steady-state flow there is no time dependence of the pore-pressure change, whereupon

the left term in equation (4.24) is eliminated.

4.2.4 van Genuchten model

For description of hydraulic behavior in the soil, va Ge u hte ’s model (van

Genuchten, 1980) was used in the modelling work done in this study (Paper B).

The important role of the hydraulic model is to properly define the degree of

saturation, as a function of suction pore-pressure head, . This relation is expressed

as:

( ) ( )[ ( | |)

] (4.25)

where is the residual degree of saturation, is the saturated degree of saturation,

is an empirical parameter describing air entry, and is an empirical parameter

related to the water extraction from the soil (Galavi, 2010). The suction pore pressure

head is defined as:

⁄ (4.26)

where is the suction pore pressure and is the weight of the water. The relative

permeability is expressed as:

( ) [ (

)

]

(4.27)

where is an empirical parameter. The hydraulic properties of a soil are often

presented in soil-water characteristic curves (SWCC). In Figure 4-6 and Figure 4-7,

SWCC’s for different values of the empirical parameters in van Genuchten´s hydraulic

model are shown.


41

Figure 4-6: The effect of the parameter (linked to air entry) on the saturation-suction relationship. (Galavi, 2010)

Figure 4-7: The effect of the parameter (linked to water extraction) on the saturation-suction. relationship (Galavi, 2010)

4.3 Parameter sensitivity

In order to evaluate to what extent the model parameters used are influencing the

computation results, parameter-sensitivity analysis might be carried out. Parameter-sensitivity analysis might be useful for minimization of computation time; i.e. by

identifying parameters being relevant for the result (significantly influencing).

Sensitivity analysis might also be used for consideration of spatial in-situ variety of soil properties (Schweiger & Peschl, 2005), temporal changes (e.g. material deterioration


42

due to loading and aging), and/or uncertainties connected to the actual quantification (e.g. laboratory tests and field tests).

In the stability-modelling performed in this study (Paper B), sensitivity analysis was

used for comparison of the three analysis approaches evaluated. The parameter

influence was investigated with respect to changed stability (FOS). For each parameter, a maximum and a minimum value were defined setting up an interval within which

the reference value for each parameter was located. The theories used are based on the

ratio of the percentage change of the result, and the percentage change of input

(creating the output change); explained in e.g. U.S. EPA (1999). The sensitivity ratio –sometimes called elasticity (e.g. U.S. EPA, 2001)—is expressed as:

( ( ) ( )

( )) (

)⁄ (4.28)

where the output values are functions, of the input values , for which no index

represents the reference input values and the indices U,L represents the changed ones;

upper/lower. In order to increase the robustness of the analysis (PLAXIS, 2012) the

sensitivity ratio was weighted by a normalized variability measure. This was done using

a sensitivity score, , defined as the ratio of the range itself (or the standard deviation)

and the reference value (U.S. EPA, 2001):

(

) (4.29)

where and defines the range limits, and is the mean (reference) value.

The total sensitivity was then calculated as:

(

∑ ( )) (4.30)

where and

are the sensitivity scores connected to the upper and lower

parameter value, respectively, and ∑ is the sum the sensitivity scores of all

parameter combinations. Five model parameters were considered. In addition to the

reference computation, one computation was run for each range boundary for each

parameter (keeping the other parameters fixed), meaning 11 computations per stability

case.

FEM-Modelling

43

5 FEM-MODELLING

5.1 Main aim

The modelling work done (Paper B) was aimed to (1) investigate potential impacts on

watercourse slopes subjected to recurring water-level fluctuations, (2) to find out

whether there are important dissimilarities between a well-established modelling

approach (based on the classical theory of saturated soil) and approaches that also

consider unsaturated soil behavior, and (3) to analyze potential parameter-influence

changes coming from the water-level changes. More details are presented in Paper B.

5.2 Strategy

The following modelling strategies were used:

Approach 1 (A1): Computations were performed using Terzaghi's definition of stress.

Flow computations were run prior to the deformation computations; the results were

then used in the deformation computations for the corresponding time step. The

consolidation computations were based on excess pore pressures.

Approach 2 (A2): For the fully coupled approach Bishop’s effective-stress equation was

used. The consolidation computations were based on the Biot’s theory of consolidation

(Biot, 1941); i.e. using total pore pressures with no distinction made between steady

state and excess pore pressures.

Approach 3 (A3): Computations were run like in A2, but with stability computations

run without consideration of negative pore pressures.

A fictive waterfront bank slope consisting of a well-graded postglacial till was simulated

to be subjected to a series of water-level-fluctuation cycles (WLFC’s). Development of

stability (in terms of factors of safety, FOS), vertical displacements, pore pressures, flow,

and model-parameter influence, was investigated for an increased number of WLFC’s.

The modelling was performed by use of the software PLAXIS 2D 2012 (PLAXIS,

2012), allowing for consideration of transient flow and performance of coupled hydro-

mechanical computations. This FEM-code was used since it is well-established, and

providing what is needed. Moreover, parameter-influence analysis was performed with


44

respect to FOS’s computed using the strength reduction technique. The parameter-

influence development with an increased number of WLFC’s, was investigated.

The bank was assumed to be infinite in the direction perpendicular to the model plane;

i.e. 2D plain-strain conditions were assumed. The initial stress situation was

determined by a Gravity loading computation; providing initial stresses generated based

on the volumetric weight of the soil (this since the non-horizontal upper model

boundary means that the K0-procedure is not suitable). Then a plastic nil-step was run,

solving existing out-of-balance forces and restoring stress equilibrium. The WLFC’s

were defined in separate calculation steps; one drawdown-rise cycle in each. Each

WLFC was followed by a FOS-computation. In A3, each FOS-computation was

preceded by a plastic nil-step; this to reset the stress situation in order to omit

consideration of negative pore pressures. Sensitivity analysis was carried out for the

three approaches. Five parameters were considered. Ranges used are presented in

Table 5-2.

5.3 Geometry, materials, and definition of changes

The model geometry was defined aimed to represent one half of a watercourse cross

section. The model dimensions and mesh conditions were defined aimed to avoid (1)

errors due to disturbances by the boundaries (due to too small model), (2) errors due to

too coarse mesh, and (3) unreasonably time consuming computations due to too fine

mesh. The slope height was set to 30 m, and the slope inclination to 1:2 (26.6°)

(Figure 5-1); both measures considered to be representative for a watercourse slope

consisting of a well-graded postglacial till, for Swedish conditions (properties presented

in Table 5-1). 15-noded triangular elements were used. The mesh density was

optimized by mesh quality checks done with the built in tool available in PLAXIS

(2012) (the average element quality was 97%). Since the area around the position

where the slope face and the external water level intersect was assumed to be most

critical, extra care was taken regarding mesh density and quality in this region.

The vertical model boundaries were set to be horizontally fixed and free to move

vertically, whereas the bottom boundary was completely fixed. The left vertical

boundary was assumed to have properties of a symmetry line, and the model bottom to

represent a material being dense and stiff relative to the soil above. Therefore, as well

the left vertical hydraulic boundary as the horizontal bottom, were defined as closed

(impermeable). The right vertical boundary was defined to be assigned pore pressures

by the head, simulating allowed in/out flow. Data-collection points were placed where

interesting changes would be expected; at the crest, down along the slope surface, and

FEM-Modelling

45

along a vertical line below the crest. The initial water-level position was set to

+22.0 m (with the level of the watercourse bottom at ± 0 m), the level-change

magnitude to 5.0 m (linearly lowered and raised), and the rate to 1.0 m/day. The

water-level changes were run continuously; i.e. without allowing pore pressures to

even out. Since the study was comparative, this was not mainly intended to reflect

frequently occurring real situations.

5.4 Models used

The soil was described using Mohr Coulomb´s model. This model was chosen for

three reasons; (1) it is well-established and widely used by engineers, (2) as soon as the

stability analysis is started (performed using the strength reduction technique), any

model defined with the same strength parameters would behave like the Mohr-

Coulomb model (i.e. stress dependent stiffness is excluded), (3) the study is

comparative whereupon differences between the modelling approaches evaluated were

focused rather than the accuracy of the soil description.

The hydraulic properties were defined using the van Genuchten´s model (van

Genuchten, 1980). The model is widely recognized and integrated in PLAXIS.

Both models were described in chapter 4.

Figure 5-1: Model geometry. Dimensions, initial water level (𝑊𝐿𝑖𝑛𝑖𝑡𝑖𝑎𝑙), and the pre-defined data-collection points (A) at the crest, and (B) at the level +22 m, are shown in the figure.


46

Table 5-1: Model input data.

Unit Value Comments

General

Drainage type - - Drained

Unsaturated weight kN/m3 19.00 Typical value

Saturated weight kN/m3 21.00 Typical value

Soil description

Young´s modulus kN/m2 40 000 2)

Poisson´s ratio - 0.30 2)

Cohesion kN/m2 8 Low, non-zero

Friction angle ° 36 Typical value

Dilatancy angle ° 3 Meeting the recommendation

Hydraulic description

Model

van Genuchten

Data set - - - Defined in order to be representative 1)

Permeability, x m/day 0.05 Found to be representative (e.g. Lafleur, 1984)

Permeability, y m/day 0.05 Found to be representative (e.g. Lafleur, 1984)

Saturated degr. of sat. - 1.0

Residual degr. of sat. - 0.026 1)

- - 1.169 Parameter linked to water extraction 1)

- - 2.490 Parameter linked to air entry 1)

- - 0 Parameter linked to pore connectivity 1) 1) The data set was manually defined by using the parameters listed. The values of the parameters ,

, , , and were chosen based on examples in Galavi (2010), comments in Gärdenäs et al. (2006), and the pre-defined data set Hypres available in PLAXIS (2012).

2) Within ranges presented in Briaud (2013); Young´s modulus for a loose glacial till (10-150 MPa), and

for a medium/compact sand (30-50 MPa), at .

NOTE: Since the result evaluation is comparative, the values of the parameters chosen are not critical.

Table 5-2: Soil parameters used in the sensitivity analysis (Min and Max), and the reference values.

E (MPa) v (-) (°) c (kPa) (°)

Min Ref Max Min Ref Max Min Ref Max Min Ref Max Min Ref Max

20 40 60 0.25 0.30 0.35 33 36 39 1 8 15 0 3 6

Results and comments

47

6 RESULTS AND COMMENTS

6.1 Evaluation of FEM-modelling approaches

6.1.1 General

More details are presented in Paper B.

6.1.2 Stability

In Figure 6-1 the failure plots from the three modelling approaches are shown;

conditions at the initial stage (Init.), after one rapid drawdown (RDD), and after one

WLFC (1); corresponding FOS-changes are presented in Figure 6-2; and FOS-

developments at further cycling are presented in Figure 6-3.

Init. RDD 1

A1

A2

A3

Figure 6-1: Failure surfaces for the initial condition, after one rapid drawdown, and after one WLFC.


48

Figure 6-2: FOS-values for initial conditions (Init.), after one rapid drawdown (RDD), and after one WLFC (1).

Figure 6-3: FOS-developme t by i reased umber o WLFC’s.

For all approaches the FOS-values were significantly decreasing due to the rapid

drawdown, slightly increasing due to the first rise, and ending up exhibiting further

increased values after 10 WLFC’s (with the initial conditions as reference values).

Lower FOS-values were obtained using the approach assuming saturated conditions

below the water-level, dry conditions above, and taking into account neither pore-

pressure changes due to deformations, nor suction (A1), than using fully coupled flow-

deformation computations, considering also suction forces (A2). Though, the lowest

FOS-values were obtained using fully coupled flow-deformation computations, but

with the negative pore pressures not considered in the stability analysis (A3).

6.1.3 Deformations, pore pressures and flow

In Figure 6-4, vertical deformations developed at the crest (point A, Figure 5-1) are

plotted. The values were registered in the middle of each WLFC, i.e. at the end of

each drawdown.

1.30

1.40

1.50

1.60

1.70

1.80

Init. RDD 1

FOS

A1

A2

A3

1.65

1.70

1.75

1.80

1.85

1 2 3 4 5 6 7 8 9 10

FOS

WLFC

A1

A2

A3


49

Figure 6-4: Vertical deformations at the crest; readings from the end of each drawdown. The reference level before the first drawdown is 0 mm.

The overall pattern is the same in all approaches; additional vertical deformations

developed with an increased number of drawdowns. The rate is decreasing although a

definitive asymptotical pattern cannot be seen.

Developments of the active pore pressure, consisting of contributions from as

well steady state as excess pore pressure, are shown below. The values were registered

in two different points; one located at the crest (point A in Figure 5-1), and one at the

initial groundwater level (point B in Figure 5-1). In Figure 6-5, the values from point

B are shown; only peak values are plotted. In Figure 6-6 the development in point A is

shown. In both plots, only values from approach A2 are presented. The trends were

similar in A3, whereas A1 is not relevant due to fully dry conditions above the water

level.

Figure 6-5: Peak-pore pressure development from A2, registered in point B at the end of each WLFC; the development in between was found to be smooth/non-fluctuating.

-24.0

-23.0

-22.0

-21.0

-20.0

-19.0

-18.0

1 2 3 4 5 6 7 8 9 10

uy

(mm

)

Number of drawdowns

A1A2A3

-8

-7

-6

-5

-4

1 2 3 4 5 6 7 8 9 10

p a

ctiv

e (k

Pa )

WLFC


50

Figure 6-6: Pore-pressure development from A2, registered in point A.

In Figure 6-7 a picture of the flow region—the area where the external water level and

the slope surface intersect—is shown. Total (resultant) flow is represented by arrows.

The picture shows the conditions prevailing after the first WLFC was run, but is

qualitatively characteristic for the endpoints of all WLFC’s.

Figure 6-7: A sample picture of the area where the external water level and the slope intersect, showing the hydraulic conditions prevailing immediately after the first WLFC run in A1. The external water level is located at +22 m whereas the groundwater table within the slope is lower. The flow directions are presented by arrows.

-80.4

-80.3

-80.2

-80.1

-80.0

-79.9

-79.8

-79.7

-79.6

0 1 2 3 4 5 6 7 8 9 10

p a

ctiv

e (k

Pa

)

WLFC


51

In Figure 6-8 - Figure 6-13 flow conditions registered immediately after each WLFC

are presented. Since the readings were taken at the end of each cycle, the external

water level and the groundwater table are almost coinciding, meaning small absolute

magnitudes of the flow values registered. The flow condition for each stage is

presented as the deviation from the average value of the 10 cycles. Maximum total

flow, and horizontal flow directed inwards (in to the slope), is presented

in Figure 6-8 and Figure 6-9, whereas maximum vertical flow (upward, , and

downward, ) is shown in Figure 6-10 and Figure 6-11. All of the maximum

and minimum values are originating from the region being marked in Figure 6-7,

except the values of . The latter ones are registered just below the groundwater

table, approximately 10 m right (inwards) from the slope surface.

Figure 6-8: Maximum total flow; average ~0.1 m/day.

Figure 6-9: Rightward directed horizontal flow; average ~0.08 m/day.

-60%

-30%

0%

30%

60%

90%

1 2 3 4 5 6 7 8 8 10

|qm

ax|(

dev

iati

on

fro

m

aver

age)

WLFC

A1

A2

A3

-100%

-50%

0%

50%

100%

150%

1 2 3 4 5 6 7 8 9 10qx,

rig

ht (

dev

iati

on

fro

m

aver

age)

WLFC

A1

A2

A3


52

Figure 6-10: Maximum upward directed flow; average ~0.05 m/day.

Figure 6-11: Downward directed flow; average ~0.05 m/day.

In Figure 6-12 and Figure 6-13, development of vertical flow, and horizontal flow,

—extracted from the same stress point in all approaches—are shown. The point is

located approximately 1 m below the initial water level, 1 m to the right from the

slope surface.

Figure 6-12: Vertical flow; average ~0.05 m/day.

-100%

-50%

0%

50%

100%

150%

200%

1 2 3 4 5 6 7 8 9 10

qy,

up (

dev

iati

on

fro

m

aver

age)

WLFC

A1

A2

A3

-100%

-50%

0%

50%

100%

150%

200%

1 2 3 4 5 6 7 8 9 10

qy,

do

wn (

dev

iati

on

fro

m

aver

age)

WLFC

A1

A2

A3

-20%

0%

20%

40%

60%

1 2 3 4 5 6 7 8 9 10

qy

(d

evia

tio

n f

rom

av

erag

e)

WLFC

A1

A2

A3


53

Figure 6-13: Horizontal flow; average ~0.05 m/day.

The flow magnitudes were found to be changing by increased number of cycles run.

Larger flow changes were captured using the fully coupled flow-deformation

computations (A2 and A3), compared to A1. The difference is particularly pronounced

for the rightward directed horizontal flow, and the downward directed vertical

flow, . Regarding and the results from A2 and A3 are to a large

extent coinciding.

6.2 Parameter influence

The sensitivity analysis performed in the study presented in Paper B did provide

parameter-influence values; total sensitivity ( ). In Figure 6-14 -values

(on the y-axis) for each of the five parameters at initial state (Init.), after one rapid

drawdown (RDD), and after one WLFC (1), are shown.

(A1) (A2) (A3)

Figure 6-14: The influence (in terms of total sensitivity) of each parameter, with respect to safety (in terms of FOS).

The stiffness parameters are influencing the stability significantly less than are the

strength parameters. The cohesion, exhibits the highest influence in all approaches;

followed by the influence of the friction angle, and the dilatancy angle, . The

-20%

0%

20%

40%

60%

1 2 3 4 5 6 7 8 9 10

qx

(d

evia

tio

n f

rom

av

erag

e)

WLFC

A1

A2

A3

0%

10%

20%

30%

40%

50%

60%

70%

ν c φ ψ E0%

10%

20%

30%

40%

50%

60%

70%

ν c φ ψ E0%

10%

20%

30%

40%

50%

60%

70%

ν c φ ψ E

Init.

RDD

1


54

influence of as well Poisson’s ratio, as of Young’s modulus, is slightly increasing

during further cycling; though remaining below 3% and 2%, respectively. Therefore,

the stiffness parameters are excluded from the presentation of the further sensitivity

development, presented in Figure 6-15 - Figure 6-17.

Figure 6-15: Sensitivity development for cohesion, c. The dotted lines are showing the initial values for each modelling approach (A10, A20, and A30).

Figure 6-16: Sensitivity development for friction angle, . The dotted lines are showing the initial values for each modelling approach (A10, A20, and A30).

50%

52%

54%

56%

58%

60%

1 2 3 4 5 6 7 8 9 10

Sen

siti

vity

(co

hes

ion

)

WLFC

A1

A2

A3

A30

A10

A20

30%

32%

34%

36%

38%

40%

1 2 3 4 5 6 7 8 9 10

Sen

siti

vity

(fr

icti

on

an

gle)

WLFC

A1

A2

A3

A30

A10

A20


55

Figure 6-17. Sensitivity development for dilatancy angle, . The dotted lines are showing the initial values for each modelling approach (A10, A20, and A30).

It is clear that the sensitivity values are depending on the stage at which the safety is

evaluated, i.e. on the numbers of WLFC’s run. For cohesion, the qualitative trend is

the same in all cases; maximum influence at drawdown and then declining with

increased number of WLFC’s. In A1 the sensitivity of cohesion gets lower than the

initial value after 5 WLFC’s; in A2 this occurs at 2 WLFC’s, and in A3 the initial value

is not reached at all during the 10 cycles. For the friction angle, the pattern is

opposite; minimum values at drawdown and then increasing. For this parameter, the

initial values are exceeded in all approaches; in A1 and A2 after 3 WLFC’s, and in A3

after 6 WLFC’s. The total change of the influence of cohesion during 10 cycles, seems

to be similar in all approaches. The increase of the friction-angle influence in A2 seems

to be fading out slower than in A1 and A3. When it comes to the dilatancy angle,

the patterns are more scattered. In all cases the sensitivity values are decreased after the

first drawdown (Figure 6-14), whereas the further development is almost constant.

Though, the absolute magnitudes (and the variations) of the dilatancy-angle influence

are very small, and further conclusions are not reasonably drawn.

5%

6%

7%

8%

9%

1 2 3 4 5 6 7 8 9 10Sen

siti

vity

(d

ilata

ncy

an

gle)

WLFC

A1

A2

A3

A30 A10 A20


56

Discussion

57

7 DISCUSSION

Concerning objective 1 (Review):

Fast growing use of non-regulated energy sources and pumped-storage hydropower

would potentially entail water levels in regulated watercourses being fluctuating more

in the future than until present. This is the view among many authors (e.g. Catrinu,

Solvang, & Korpås, 2010; Whittingham, 2008; Connolly, 2010; Zuber, 2011). Since

production of floating debris due to bank erosion and slope failure is a current matter

(e.g. Åstrand & Johansson, 2011), an increased degree of fluctuating water level will

probably worsen the situation in locations being exposed. This hypothesis is

substantiated by statements made in K. Zhang et al. (2004), speaking of varied mean

water levels—changed water-level ranges within which the short-term variations are

occurring—as enablers of erosion. Such changes could directly mean reduced slope

stability.

There is a lack of studies that actually evaluate recurring water-level variations.

Though, there have been a few addressing issues connected to drawdown (e.g. Yang et

al., 2010; Pinyol, Alonso, Corominas, & Moya, 2011), and rise (e.g. Jia et al., 2009;

Cojean & Caï, 2011). Thus, it is reasonable to assume that at least the issues noticed for

single drawdowns and rises (pore-pressure gaps, suction loss, effects of rapidly increased

water pressures in pores and discontinuities, harmful flow patterns, soil material

migration, retrogressive failure development etc.), also would be faced as a

consequence of many changes being cyclic recurring.

Moving water, altering hydraulic gradients, and cyclic loading, are all factors

potentially entailing soil degradation and changed soil properties. This in turn could

mean strength reduction and stability issues (J. M. Duncan, 2005). When it comes to

proper description of soils, there seems to be factors that could be incorporated in

modelling for improved accuracy. For instance, particle shape for consideration of

degradation potential (Cho, 2006; JM Rodriguez, 2013), and creep for more detailed

description of long-term deformations (Tatsuoka, 2007).

In studies addressing flow modeling issues, soil material characteristics are only in

exceptional cases specified. Also when fundamental soil characteristics are fairly

described, like in e.g. Yang et al. (2010), strength/stiffness properties are rarely


58

considered. Therefore, the coupling between soil deformation, pore-pressure

development, and water flow conditions are often overlooked and/or missed.

Given the important and critical connections involving pore pressures, soil strength,

soil-deformation development, (and so on)—stated in Huang & Jia (2009), Galavi

(2010), and Sheng (2011)—it seems to be unreasonable to use simple limit-

equilibrium methods for analysis of slopes subjected to water-level fluctuations.

Concerning objective 2 (Impacts on a slope subjected to water-level

fluctuations—FEM-modelling):

The modelling performed in this study—given the specific conditions of soil

properties, slope geometry, and pattern of water-level changes—showed that the

water-level fluctuations resulted in growing stability and development of vertical

deformations at the slope crest. The stability increase is to assign to the fact that the

groundwater table is dropping (remained at a lower level) for each WLFC. Though,

since rapid changes of pore pressures and flow rates were noticed within the slope,

other upcoming issues could probably occur as well suddenly as in the long term.

The development of the pore-pressure peak values (registered at the initial water level,

point B in Figure 5-1) shows how the pore pressure at the end of each WLFC, slowly

becomes more and more negative. This increase of the negative pore pressure indicates

that the groundwater table is dropping for each cycle, contributing to an increased

stability. This in turn means that equilibrium between groundwater level and external

water level was not reached during the ten cycles. If such fluctuations are taking place

for a long time, negative pore pressures could falsely remain high. If then the

fluctuation stops, the water levels would go back to equilibrium, and the extra shear

strength would be lost. This phenomenon of suction loss would potentially get

consequences similar to those noticed in situations of wetting-induced strength loss

presented in e.g. (Jia et al. (2009). The pore-pressure development registered both in

point A and B did show asymptotic patterns. Also the deformation increase was fading

out. This might indicate that the slope was approaching a state where only elastic

strains were developed at further WLFC’s.

In situations where internal erosion is taking place, the soil is definitely getting changed

properties. Since e.g. hydraulic conductivity has been shown to significantly affect the

stability of reservoir slopes consisting of sand and silt (Liao et al., 2005), material

changes would be preferably considered in long-term modelling.

Despite the fact that the absolute magnitudes of differences identified in this study were

small—FOS-values, pore pressures, flows etc.—these do nonetheless demonstrate

Discussion

59

important dissimilarities concerning the ability to capture/simulate real soil-water

interactions and changes.

Further investigations of processes taking place within slopes being subjected to

recurrent WLF’s, would desirably be conducted. For instance by laboratory/scale tests

like what was presented in Jia et al. (2009), combined with moelling.

The results obtained in this study are reflecting effects occurred under the specified

conditions of hydraulic conductivity, rate of water-level change, and slope geometry.

Since these factors are unquestionably affecting processes taking place within a

watercourse slope, it would be reasonable to include variation also of such non-

constitutive parameters in further studies.

Concerning objective 3 (Evaluation of FEM-modelling approaches):

Lower FOS-values were obtained using the classical approach, A1 than those obtained

using approach A2. This could be reasonably explained by the consideration taken to

the contribution of suction forces to the stability in A2; i.e. due to higher effective

stresses.

The fact that the lowest FOS’s were obtained by using A3, would suggests that this

approach was most conservative in this study, in the sense of not overestimating the

stability. This is unexpected since suction was not considered at all in A1, whereas

suction was considered throughout the entire cycle in A3, until the FOS-computation

was run. Anyhow, whether the unexpected result is connected to the nil-steps

computed in A3 done in order to neglected suction, or to unrealistic computation of

the soil weight above the water level, the FOS result of A3 is to be treated with

caution.

Use of A1 resulted in smaller vertical crest deformations compared to the other

approaches. This suggests that the two-way interaction between pore pressure changes

and deformation development in the fully coupled computations (A2 and A3), could

capture conditions that bring larger deformations. This means a potential

underestimation in A1. Moreover, the flow developments registered in A2 and A3

were significantly more fluctuating than those in A1; as well horizontally as vertically.

The difference was seen to be particularly pronounced for the rightward directed

horizontal flow, and the downward directed vertical flow. In addition, the fluctuating

negative pore pressure registered at the crest does underline that central information

about processes taking place in the slope is well captured in A2. Since changes such as

those mentioned in turn are governing as well water-transport (e.g. efficiency of

dissipation of excess pore pressures), as soil-material transport (i.e. susceptibility to


60

internal erosion to be initiated and/or continued), modelling approaches not being

fully coupled are probably not suitable for analysis of processes being governed by

fluctuating flows, altering hydraulic gradients, unloading/reloading etc. In that sense—

in agreement with what was concluded in Sheng et al. (2013)—there seems to be

situations where approaches considering unsaturated soil behavior is not necessarily less

conservative than are classical ones.

Since hydraulic models do rely on empirical soil-specific parameters, there are potential

uncertainties connected to the level of accuracy at which these have been quantified.

Still, this uncertainty is to be related to the extra information possibly obtained at

consideration of specific features of unsaturated soils.

Concerning objective 4 (Parameter-sensitivity analysis):

The parameter-sensitivity analysis results obtained using A1 seems to be quantitatively

more comparable to those obtained using A3 than to those obtained using A2. This

suggests that consideration of suction at stability computation (only true in A2), does

directly affect the parameter influence on the results. However, major differences were

not found. Since the sensitivity was computed with respect to FOS determined by

using the strength reduction technique, the low and insignificantly changing sensitivity

values of the stiffness parameters were expected. The non-constant sensitivity values of

the dilatancy angle, might indicate that the occurrences of plastic strains are somewhat

different in the different stages.

It might be useful to consider parameter influence for evaluation of the accuracy

needed for definition of model input data; as well absolute magnitudes, as the

development patterns. Moreover, when making up modelling/design strategies, parameter changes (e.g. by time or by external changes taking place), could be valuably

included.

Conclusions

61

8 CONCLUSIONS

8.1 General

Concerning objective 1 (Review):

It seems to be consensus among researchers and experts that the fast growing use

of non-regulated energy sources and pumped-storage hydropower, will entail

water levels in regulated watercourses being fluctuating more in the future than

until present.

Water-level changes have been shown to significantly influence slope stability;

failure due to suction loss, effects of rapidly increased water pressures (as well in

pores as in discontinuities), potentially harmful flow patterns, internal erosion,

and retrogressive failure development.

Moving water, altering hydraulic gradients, soil-material transport, and cyclic

external loading, are all factors potentially entailing soil degradation, property

changes, strength reduction, and stability issues.

Given the important and critical connections involving pore pressures, soil

strength, and soil-deformation development, it is not reasonable to use simple

limit-equilibrium methods for analysis of slopes subjected to water-level

fluctuations.

Concerning objective 2 (Impacts on a slope subjected to water-level

fluctuations—FEM-modelling):

Water-level fluctuations resulted in growing stability, development of vertical

deformations at the slope crest, and rapidly fluctuating flows (as well horizontal

as vertical) within the slope.

Even though the stability might remain unchanged or even increase due to

fluctuations of an external water level, rapid changes of pore pressures and flow

within the slope, could mean upcoming issues as well suddenly as in the long

term.

Concerning objective 3 (Evaluation of FEM-modelling approaches):

Modelling assuming strictly saturated or dry conditions resulted in—restricted to

slope stability in terms of safety factors—higher conservatism compared to


62

modelling considering also the behavior of unsaturated soil and fully hydro-

mechanical coupling.

Modelling considering the behavior of unsaturated soil and fully hydro-

mechanical coupling, but with suction neglected at stability computation, was

shown to give safety factors even lower than did classical modelling. Whether

this was due to the saturation-suction relation in the hydraulic model used, or

due to the nil-steps run prior to the FOS-computations, the importance of

being careful when complex situations are designed, is however underlined.

Advanced modelling seems to allow for rapid changes of pore pressures and

flow to be more realistically described than do classical modelling. Since such

changes in turn are governing as well water-transport (i.e. the efficiency of

dissipation of excess pore pressures), as soil-material transport (i.e. susceptibility

to internal erosion to be initiated and/or continued), simplified modelling

approaches are not suitable for analysis of processes being governed by

fluctuating flows, altering hydraulic gradients, unloading/reloading etc.

Concerning objective 4 (Parameter-sensitivity analysis):

The influence of the parameters on the slope stability was found to be changing

as a result of WLFC’s taking place.

For evaluation of the accuracy needed for modelling-input data, it might be

useful to consider parameter influence; as well total values, as the development

patterns. Moreover, when making up modelling/design strategies, inclusion of

parameter changes (e.g. by time or by external changes taking place), could be

valuable.

8.2 To be further considered

In order to properly simulate groundwater flow—capturing also effects besides critical

development of pore-pressures, flow patterns, and subsequent stability changes—consideration should be taken to:

Use of a more advanced constitutive soil model (e.g. the Hardening soil model

PLAXIS, 2012) would potentially reduce inaccuracy coming from e.g.

improper description of plastic deformations and stiffness changes. The water-

level fluctuation effects would also be investigated using models based on the

Basic Barcelona Model (Alonso, Gens, & Josa, 1990).

Incorporate varied hydraulic conductivity in sensitivity analyses.

Conclusions

63

Vary frequencies and rates of water-level changes, slope geometries etc. (in

previous studies simple approaches are used focusing on one or a few cycles), for

further investigation.

Design a modelling approach for application to long-term effects of WLF’s on

potential debris production along regulated watercourses.

Indirect consider internal-erosion in embankments (dams and/or watercourse

banks), by incorporating parameter changes in the modelling.

Further investigate processes taking place within slopes being subjected to

recurrent WLF’s, e.g. by laboratory/scale tests like what was presented in Jia et

al. (2009) .


64

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65

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Paper A

Effects of External Water-Level Fluctuations on Slope

Stability

J. M. Johansson and T. Edeskär

Electronic Journal of Geotechnical Engineering vol. 19, no. K, pp. 2437–2463, 2014.

Jens M. A. Johansson

Tommy Edeskär

ABSTRACT

KEYWORDS:

- 2437 -

Vol. 19 [2014], Bund. K 2438

INTRODUCTION

Vol. 19 [2014], Bund. K 2439

WATER LEVEL VARIATION – MODES AND SOURCES

General

Long-term natural variations

Vol. 19 [2014], Bund. K 2440

Short-term natural variations

Figure 1:

Vol. 19 [2014], Bund. K 2441

Watercourse regulation

Figure 2:

Vol. 19 [2014], Bund. K 2442

WATER-LEVEL VARIATION AND SLOPE STABILITY – EFFECTS AND CONSIDERATIONS

Analysis of slope stability

Vol. 19 [2014], Bund. K 2443

= ´= = + ( ) tan ( )

Vol. 19 [2014], Bund. K 2444

Figure 3:

Vol. 19 [2014], Bund. K 2445

Vol. 19 [2014], Bund. K 2446

Water-level drawdown

Vol. 19 [2014], Bund. K 2447

Vol. 19 [2014], Bund. K 2448

Water-level rise

Figure 4

Vol. 19 [2014], Bund. K 2449

Water-level fluctuation: drawdown-rise combinations

Hydrological modeling

Vol. 19 [2014], Bund. K 2450

Vol. 19 [2014], Bund. K 2451

Seepage – internal erosion

Vol. 19 [2014], Bund. K 2452

Vol. 19 [2014], Bund. K 2453

Vol. 19 [2014], Bund. K 2454

CONCLUDING DISCUSSION

Vol. 19 [2014], Bund. K 2455

Vol. 19 [2014], Bund. K 2456

ASPECTS TO BE FURTHER CONSIDERED

Vol. 19 [2014], Bund. K 2457

ACKNOWLEDGEMENTS

REFERENCES

Vol. 19 [2014], Bund. K 2458

Vol. 19 [2014], Bund. K 2459

Vol. 19 [2014], Bund. K 2460

Vol. 19 [2014], Bund. K 2461

Vol. 19 [2014], Bund. K 2462

Vol. 19 [2014], Bund. K 2463

Paper B

Modelling approaches considering impacts of water-level

fluctuations on slope stability

J. M. Johansson and T. Edeskär

To be submitted

1

Modelling approaches considering impacts of water-level fluctuations

on slope stability

Jens M.A. Johansson1; Tommy Edeskär

2

Abstract

Waterfront slopes are affected by water-level fluctuations originating from as well natural sources (e.g. tides and wind

waves), as non-natural sources such as watercourse regulation involving daily or hourly recurring water-level

fluctuations. Potentially instable slopes in populated areas means risks for as well property as human lives. In this

study, three different approaches used for hydro-mechanical coupling in FEM-modelling of slope stability, have been

evaluated. A fictive slope consisting of a till-like soil material has been modeled to be exposed to a series of water-

level fluctuation cycles (WLFC’s). Modelling based on assuming fully saturated conditions, and with computations of

flow and deformations separately run, has been put against two approaches being more sophisticated, with

unsaturated-soil behavior considered and with computations of pore-pressures and deformations simultaneously run.

Development of stability, vertical displacements, pore pressures, flow, and model-parameter influence, has been

investigated for an increased number of WLFC’s. It was found that more advanced approaches did allow for capturing

larger variations of flow and pore pressures. Classical modelling resulted in smaller vertical displacements, and

smoother development of pore-pressure and flow. Flow patterns, changes of soil density (linked to volume changes

governed by suction fluctuations), and changes of hydraulic conductivity, are all factors governing as well water-

transport (e.g. efficiency of dissipation of excess pore pressures) as soil-material transport (i.e. susceptibility to

internal erosion to be initiated and/or continued). Therefore, the results shown underline potential strengths of

sophisticated modelling. Parameter influence was shown to change during water-level cycling.

Key words: Hydro-mechanical coupling; Water-level fluctuations; Slope stability; Unsaturated soil; FEM-analysis;

Parameter influence

1 Introduction

When it comes to recurring water-level fluctuations in watercourses and reservoirs, effects of hydrological

changes on slope stability are often overlooked. There is a predominance of studies focused on

bioenvironmental issues [1]. The activity of watercourse regulation for water storage, enabling irrigation,

freshwater provision, and hydropower production, often means occurrence of water-level fluctuations (e.g.

[2–4]). The growing use of non-regulated energy sources (e.g. wind and solar) [5, 2, 6, 7]—bringing

significant emphasis put on wind-power development and exploitation—entails a need of energy balancing

by water storage in reservoirs. According to [8], this will necessarily require that flow magnitudes and

water level heights, to a greater extent than until present, will vary in the future. It is highly expected to get

increased variations of the reservoir water levels; hour to hour, day to day and/or seasonally [9, 2, 4].

The process of rapid drawdown is described and investigated by many authors e.g. [10–13]. Also water-

level rise might cause problems. In studies concerning the Three Gorges Dam site (e.g. [14]), slope stability

reductions registered during periods of water-level rise, have been reported. Rise of water levels has been

shown to cause stress redistributions due to external loading, wetting induced loss of negative pore

pressures, and seepage effects. Subsequently, these changes have been shown to potentially cause loss of

shear strength, soil structure collapse, and development of settlements and/or slope failure [15]. It has been

emphasized that a delayed change of pore water pressure inside a slope—relative to the external water

level—may result in significant movements of water within the slope; creating seepage forces being

adversely affecting the stability [15].

Despite well-known limitations of limit-equilibrium methods (LEM) (e.g. [11, 16]), these are well

established and widely used. For capturing important relations between pore pressure, soil strength, and

1 Department of Civil, Environmental and Natural Resources Engineering, Luleå University of Technology 2 Department of Civil, Environmental and Natural Resources Engineering, Luleå University of Technology

2

soil-deformation development, the use of finite-element methods (FEM) have been more and more spread.

Long-term perspectives are important for consideration of altering hydraulic gradients, stress-strain

changes, and development of influenced waterfront zones. Still, reliable relationships between soil

deformation, pore-pressure development, and unsteady water-flow conditions are often deliberately

overlooked or incidentally missed [1]. Though, improved accessibility of high computer capacity allows for

more and more advanced analyses to be performed. In addition, optimized designs and constructions are

increasingly demanded and less conservative design approaches are therefore often desired. This is not at

least linked to economic and environmental aspects. One non-conservative view in slope-stability analysis

regards consideration of negative pore pressures in unsaturated soils. Taking into account negative pore

pressures is generally associated with counting on extra contributions to the shear strength of the soil,

resulting in extra slope stability. The well-known and widely used Terzaghi´s equation of effective-stress

( – u) is simply based upon the difference between the total stress, and the pore pressure, u. The

equation implies that there are only two possible soil conditions; (1) presence of pore pressure coming from

fully water filled pores, and (2) no pore pressure. The validity of this expression is widely accepted in

saturated soil mechanics, but has since a long time been found to not be applicable for partly saturated

conditions, e.g. [17]. The possibility to control the mechanical response of unsaturated soils by considering

two stress-state variables—(1) net stress, i.e. the difference between the total stress and the pore-air

pressure, and (2) suction, i.e. the difference between the pore-air pressure and the pore-water pressure—was

firstly demonstrated by [18]. Though, the earliest effective-stress equation for consideration of a varying

degree of saturation and suction was presented by Bishop in the end of the 1950th. For description of the

amount of water being present in unsaturated soils, soil-water characteristic curves (SWCC) are important.

The degree of saturation, (among other alternatively use quantities) appears to most closely control the

behavior of unsaturated soil [19]. For proper consideration of the behavior of unsaturated soils, fully

coupled hydro-mechanical computations are needed (e.g. [20–22]).

Among studies with hydro-mechanical approaches applied, many are addressing sediment-loading

problems; sources, changed river shapes caused by sedimentation etc. (e.g. [23–25]). Consequently, the

slope-stability part is often either absent or considered by using LEM-approaches (e.g. [26–28]). There are

a few studies addressing effects of large-scale reservoir water-level fluctuations on slope stability; e.g. [29]

where saturated-unsaturated seepage analysis was combined with LEM-stability analysis, and [30] where

evaluation of the approach of analyzing stability using critical slip fields. The complexity of how factors

such as drawdown rate, permeability, and slope height actually do influence the stability of a slope

subjected to drawdown, was discussed in [20].

In [31] it was emphasized the misconception concerning that analysis approaches considering principles

of unsaturated soil mechanics are always less conservative than are classical ones. It was stated that the

overall subject is at its early stage, that there are numerous areas in which practical application of the

principles of unsaturated soil mechanics are central, and that as well experimental as theoretical advances

are expected.

The present study is aimed to evaluate results from three approaches of hydro-mechanical coupling in

FEM-modelling of slope stability; (1) semi-coupled modelling, (2) fully coupled modelling, and (3) fully

coupled modelling, but with negative pore pressures neglected in stability computations. In (1) flow

computations were run prior to those of deformations, and Terzaghi’s effective-stress definition was used.

In (2) pore-pressure development and deformations were simultaneously computed, utilizing Bishop’s

effective-stress definition, considering also the behavior of unsaturated soils. Approach (3) is similar to (2),

but with suction effects in stability computation neglected. A fictive waterfront bank slope consisting of a

well-graded postglacial till, typical for Swedish conditions, was modeled to be subjected to a series of

water-level-fluctuation cycles (WLFC’s). Development of stability (in terms of factors of safety, FOS),

vertical displacements, pore pressures, flow, and model-parameter influence, was investigated for an

increased number of WLFC’s. The modelling was performed by use of the software PLAXIS 2D 2012 [32],

allowing for consideration of transient flow and performance of coupled hydro-mechanical computations.

Moreover, parameter-influence analysis was performed with respect to FOS’s calculated using the strength

reduction technique; i.e. the parameter-influence development with an increased number of WLFC’s was

investigated.

3

2 Methodology

2.1 Model design and preparations

2.1.1 Geometry, materials, and definition of changes

The model geometry was defined aimed to represent one half of a watercourse cross section. The slope

height was set to 30 m, and the slope inclination to 1:2 (26.6°) (Figure 2-1); both measures considered to be

representative for a watercourse slope consisting of a well-graded till, typical for Swedish conditions

(properties presented in Table 1). 15-noded triangular elements were used.

The vertical model boundaries were set to be horizontally fixed and free to move vertically, whereas the

bottom boundary was completely fixed. The left vertical boundary was assumed to have properties of a

symmetry line, and the model bottom to represent a material being dense and stiff relative to the soil above.

Therefore, as well the left vertical hydraulic boundary as the horizontal bottom, were defined as closed

(impermeable). The right vertical boundary was defined to be assigned pore pressures by the head,

simulating allowed in/out flow. Data-collection points were placed where interesting changes would be

expected; at the crest, down along the slope surface, and along a vertical line below the crest.

The initial water-level position was set to +22.0 m (with the level of the watercourse bottom at ± 0 m), the

level-change magnitude to 5.0 m (linearly lowered and raised), and the rate to 1.0 m/day. In [33] situations

with daily magnitudes of water-level variations in pumped-storage hydropower (PSH) in the order of 10 m

were discussed, and in [20] a rate of 1 m/day was used in rapid drawdown analysis. Since this study is

comparative, the rate used was chosen in order to create a stability reduction at drawdown, avoid failure,

and still be within a relevant and representative range. The water-level changes were run continuously; i.e.

without allowing pore pressures to even out.

Figure 2-1: Model geometry. Dimensions, initial water level (WLinitial), and the pre-defined data-collection

points—one at the crest (A) and one at the level +22 m (B)—are shown in the figure.

4

Table 1: Model input data used.

Unit Value Comments

General Drainage type - - Drained

Unsaturated weight kN/m3 19.00 Typical value

Saturated weight kN/m3 21.00 Typical value

Soil description

Young´s modulus kN/m2 40 000 2)

Poisson´s ratio - 0.30 2)

Cohesion kN/m2 8 Low, non-zero

Friction angle ° 36 Typical value

Dilatancy angle ° 3 Meeting the recommendation

Hydraulic description

Model

van Genuchten

Data set - - - Defined in order to be representative 1)

Permeability, x m/day 0.05 Found to be representative (e.g. Lafleur, 1984)

Permeability, y m/day 0.05 Found to be representative (e.g. Lafleur, 1984)

Saturated degr. of sat. - 1.0

Residual degr. of sat. - 0.026 1)

- - 1.169 Parameter linked to water extraction 1)

- - 2.490 Parameter linked to air entry 1)

- - 0 Parameter linked to pore connectivity 1)

1)

The data set was manually defined by using the parameters listed. The values of the parameters , , , ,

and were chosen based on examples in [22], comments in [35], and the pre-defined data set Hypres available in

PLAXIS 2D 2012 [36]. 2)

Within ranges presented in [37]; Young´s modulus for a loose glacial till (10-150 MPa), and for a medium/compact

sand (30-50 MPa), at - .

NOTE: Since the result evaluation is comparative, the values of the parameters chosen are not critical.

2.2 Modelling strategy

2.2.1 Approaches evaluated

Approach 1 (A1):

The computations were performed using Terzaghi's definition of stress, i.e. [ ], where is the

normal stress, is the effective normal stress, and is the pore pressure consisting of steady-state pore

pressure (from the phreatic level or groundwater flow computations) and excess pore pressure (coming

from undrained behavior or consolidation; i.e. temporal pore-pressure changes governed by the WLF’s and

captured by consideration of transient water flow). In this semi-coupled approach, flow computations were

run prior to the deformation computations. Pore pressures from the flow computations were, for each time

step, used in the deformation computation for the corresponding time step. The consolidation computations

were based on excess pore pressures.

Approach 2 (A2): For the fully coupled approach Bishop’s effective-stress equation was used. The consolidation

computations were based on the Biot’s theory of consolidation [38]; i.e. using total pore pressures with no

distinction made between steady state and excess pore pressures.

Approach 3 (A3):

Computations run like in A2, but with stability computations run without consideration of negative pore

pressures.

5

2.2.2 Behavior of unsaturated soil

For the fully coupled approaches (A2 and A3), Bishop’s effective-stress equation was used, allowing for

capturing the unsaturated soil behavior:

( ) (2.1)

where is the pore-air pressure, the pore-water pressure, and the experimentally determined matric-

suction coefficient being related to the degree of saturation, . The experimental evidences of are sparse

[22], and the role of the parameter has been generally questioned [19]. In PLAXIS 2D 2012 the matric-

suction coefficient is substituted with the effective degree of saturation, , defined as:

( ) ( ) (2.2)

where is the degree of saturation and the indices max and min denote maximum and minimum values.

Moreover, in this study the air-pore pressure is assumed to be negligibly low, whereupon the effective

stress is simply expressed as:

(2.3)

The effective degree of saturation is used also for definition of actual soil unit weight, :

( ) (2.4)

where the indices denote unsaturated and saturated conditions.

2.2.3 Models used

The constitutive behavior of the soil was described by Mohr-Coulomb’s elastic-perfectly plastic model.

This model was chosen for three reasons; (1) it is well-established and widely used by engineers, (2) as

soon as the stability analysis is started (performed using the strength reduction technique), any model

defined with the same strength parameters would behave like the Mohr-Coulomb model (i.e. stress

dependent stiffness is excluded), (3) the study is comparative whereupon differences between the modelling

approaches evaluated are focused rather than the accuracy of the soil description. For other

permissions/claims, models like e.g. the Hardening soil model [32] (allowing for description of more

realistic elastic-plastic behaviors), or the Basic Barcelona Model [39] (specifically developed for

description of the stress-strain behavior in unsaturated soils), could have been used. The soil was defined by

the variables listed in Table 1. The elastic behavior is described by the parameters Young´s modulus, and

Poisson´s ratio, . The full three-dimensional yield criterion consists of six yield functions; two functions in

each plane of stress pairs. Each yield function depends on one stress pair (one combination of the principal

effective stresses), the friction angle, and the cohesion, . These are defining the ultimate state up to

which the strains are elastic; for stress conditions entailing that the yield function is reached, the strain

response becomes perfectly plastic. For these conditions the strain magnitudes are determined using flow

rules based on plastic potential functions. These are—similar to the yield functions—depending on the

stress state, but are consisting of three plastic parameters; in addition to the friction angle, and the

cohesion, , also the dilatancy angle, . This parameter is used for description of additional resistance

against shearing, originating from dilatant behavior.

For description of hydraulic behavior in the soil, van Genuchten’s model [40] was used. The important

role of the hydraulic model is to properly define the degree of saturation, as a function of suction pore-

pressure head, . This relation, often presented in soil-water characteristic curves (SWCC), is expressed:

( ) ( )[ ( | |)

] (2.5)

6

where is the residual degree of saturation, is the saturated degree of saturation, is an empirical

parameter describing air entry, and is an empirical parameter related to the water extraction from the

soil. The relative permeability, ( ) is expressed as:

( ) [ (

)

]

(2.6)

where is an empirical parameter. The relative permeability is in the flow computation reducing the

permeability according to ( ⁄ ) where is the saturated permeability.

2.2.4 Factor of safety

The strength reduction (or phi/c-reduction) technique is based on FOS’s defined as the ratio between the

available shear strength and the shear stress at failure. In PLAXIS 2D 2012 this is mathematically handled

by expressing ratios of the strength parameters:

(2.7)

where is a controlling multiplier representing the safety factor; c is the cohesion; is the friction

angle; and the index reduced indicates values being successively decreased until failure occurs. For

indication of global instability of soil slopes, the non-convergence of solutions is often used. This is

indicated by the condition when a stress state at which the failure criterion and the global equilibrium can

be simultaneously satisfied, is not found [10].

2.2.5 Parameter-sensitivity analysis

The parameter influence was investigated with respect to changed FOS. For each parameter, a maximum

and a minimum value were defined spanning a representative interval within which the reference value for

each parameter (presented in Table 1) was located. The theories used are based on the ratio of the

percentage change of the result, and the percentage change of input (creating the output change); explained

in e.g. [41]. The sensitivity ratio [42] is expressed as:

( ( ) ( )

( )) (

)⁄ (2.8)

where the output values are functions, of the input values, for which no index represents the reference

input values and the index U,L represents the changed ones; upper/lower. In order to increase the

robustness [32] of the analysis, the sensitivity ratio was weighted by a normalized variability measure. This

was done using a sensitivity score, , defined as the ratio of the range itself and the variable mean value

[42]:

(

) (2.9)

where and define the range limits, and is the mean (reference) value. The total sensitivity

was then calculated as:

7

(

∑ ( )) (2.10)

where and

are the sensitivity scores connected to the upper and lower parameter value, respectively,

and ∑ is the sum the sensitivity scores of all parameter combinations. Five model parameters were

considered in the analysis; the variation boundaries are presented in Table 2. For each parameter, one

computation was run for each range boundary, keeping the other parameters fixed. This was giving the

sensitivity ratios (eq. 2.8) and the sensitivity scores (eq. 2.9). The total scores (eq. 2.10) were then plotted.

Table 2: Soil parameters used in the sensitivity analysis (Min and Max), and the reference values.

E (MPa) v (-) (°) c (kPa) (°)

Min Ref Max Min Ref Max Min Ref Max Min Ref Max Min Ref Max

20 40 60 0.25 0.30 0.35 33 36 39 1 8 15 0 3 6

2.3 Computations

The computations were performed in PLAXIS 2D 2012 [32]. This FEM-code was used since it is well-

established and providing what was needed. The bank was assumed to be infinite in the direction

perpendicular to the model plane; i.e. 2D plain-strain conditions were assumed. A1 was performed in

Classical calculation mode whereas A2 and A3 were run in Advanced mode. The initial stress situation was

determined by a Gravity loading phase; providing initial stresses generated based on the volumetric weight

of the soil (this since the non-horizontal upper model boundary means that the K0-procedure is not

suitable). Then a plastic nil-step was run, solving existing out-of-balance forces and restoring stress

equilibrium. The WLFC’s were defined in separate calculation steps; one drawdown-rise cycle in each.

Each WLFC was followed by a FOS-computation. In A3, each FOS-computation was preceded by a plastic

nil-step; this to restore the stress equilibrium in order to omit consideration of negative pore pressures.

Sensitivity analyses were carried out for the three approaches.

3 Results and comments

3.1 Stability

In Figure 3-1 the failure plots from A1 are shown; conditions at the initial stage (Init.), after one rapid

drawdown (RDD), and after one WLFC (1). Fully developed failure surfaces were found for all WLFC’s in

all of the approaches.

Init. RDD 1

Figure 3-1: Failure surfaces for the initial condition, after one rapid drawdown, and after one WLFC, for

approach A1.

The changed failure-surface geometry—moved upward at drawdown—shows the supporting effect of the

external water level. Since the slope was stable also after the drawdown, the reference parameter values and

geometry used was found to be suitable for further analysis. The FOS-development is presented in Figure

3-2.

8

Figure 3-2: FOS-values for initial conditions (Init.), after one rapid drawdown (RDD), and after one

WLFC (1) (picture to the left); and development by additional WLFC’s (picture to the right).

For all approaches—with the initial conditions as references—the FOS-values were significantly decreasing

due to the rapid drawdown (about 17% for A1, and 16% for A2 and A3), slightly increasing due to the first

WLFC (3% for A1 and 2% for A2 and A3), and ending up exhibiting further increased values after 10

WLFC’s (a total increase of 6 % for A1, and 5% for A2 and A3). The results show that the stability growth

continues during the entire period studied.

Lower FOS-values were obtained using the approach assuming saturated conditions below the water-

level, dry conditions above, and taking into account neither pore-pressure changes due to deformations, nor

suction (A1), than using fully coupled flow-deformation computations, considering also suction forces

(A2). This is reasonably explained by the consideration taken to the contribution of suction forces to the

stability. The lowest FOS-values were obtained in A3; i.e. fully coupled flow-deformation computations,

but with the negative pore pressures not considered in the stability analysis. These FOS-values were found

to be even lower than those obtained in A1; this although suction was neglected also in A1.

3.2 Deformations, pore pressures and flow

In Figure 3-3, vertical deformations developed at the crest (point A, see Figure 2-1) are presented. The

values were registered in the middle of each WLFC, i.e. at the end of each drawdown.

Figure 3-3: Vertical deformations at the crest; readings from the end of each drawdown. The reference

level before the first drawdown is 0 mm.

The overall pattern is the same in all approaches; additional vertical deformations are developed with an

increased number of drawdowns. The rate is decreasing although a definitive asymptotical pattern cannot

be seen. Utilization of A1 seems to give less deformations developed compared to both A2 and A3. This

was found to be true at all depths vertically below point A (see Figure 2-1).

The development of the active pore pressure, (consisting of contributions from as well steady

state as excess pore pressure) measured in A2, is shown in Figure 3-4. The developments were registered in

two different points; one located at the crest (point A in Figure 2-1), and one at the initial groundwater level

(point B in Figure 2-1).

1.30

1.40

1.50

1.60

1.70

1.80

Init. RDD 1

FOS A1

A2

A3 1.65

1.70

1.75

1.80

1.85

1 2 3 4 5 6 7 8 9 10

FOS

WLFC

A1A2A3

-24.0-23.0-22.0-21.0-20.0-19.0-18.0

1 2 3 4 5 6 7 8 9 10

uy

(mm

)

Number of drawdowns

A1

A2

A3

9

Figure 3-4: Pore-pressure development from A2, registered in point B (picture to the left), and in point A

(picture to the right).

For point B only peak-values are presented, registered at the end of each WLFC. The development in

between was found to be smooth/non-fluctuating. The development of the peak values shows how the

active pore pressure—at the end of each WLFC—slowly becomes more and more negative; an increase

exhibiting a higher rate between 1 and 5 WLFC’s than for further cycling, seeming to be asymptotically

fading out. The increase of the negative pore pressure indicates that the groundwater table is dropping for

each cycle. To the right, suction effects above the water level is clearly seen. Also in this plot there is a

fading out pattern.

In Figure 3-5 a typical picture of the flow region—the area where the external water level and the slope

surface intersect—is shown. The external water level is back to the level +22, immediately after one WLFC

has been run, whereas the groundwater table is lagging behind.

Figure 3-5: A sample picture of the area where the external water level and the slope intersect, showing the

hydraulic conditions prevailing immediately after the first WLFC run in A1. The external water level is

located at +22 m whereas the groundwater table within the slope is lower. The flow directions are

presented by arrows.

In Figure 3-6, Figure 3-7 and Figure 3-8 flow conditions immediately after each WLFC are plotted. Since

the readings were taken at the end of each cycle, the external water level and the groundwater table are

almost coinciding, meaning small absolute magnitudes of the flow values registered. The flow condition for

each stage is presented as the deviation from the average value of the 10 cycles. Maximum total flow,

and horizontal flow directed inwards (in to the slope), is presented in Figure 3-6, whereas

maximum vertical flow (upward directed, , and downward directed, ) is shown in Figure 3-7.

All of the maximum and minimum values are originating from the region being marked in Figure 3-5,

except the –values. The latter ones are registered just below the groundwater table, approximately

10 m right (inwards) from the slope surface.

-8

-7

-6

-5

-4

1 2 3 4 5 6 7 8 9 10

p a

ctiv

e (

kPa

)

WLFC

-80.4

-80.2

-80.0

-79.8

-79.6

0 1 2 3 4 5 6 7 8 9 10

p a

ctiv

e (k

Pa

)

WLFC

10

Figure 3-6: Maximum total flow in the slope; average ~0.1 m/day (picture to the left), and rightward

directed horizontal flow; average ~ 0.08 m/day (picture to the right).

Figure 3-7: Maximum upward directed flow (picture to the left), and downward directed flow (picture to

the right); both averages ~0.05 m/day.

The flow magnitudes were found to change from one stage to another; i.e. by increased number of cycles.

Larger flow changes were captured using the fully coupled flow-deformation computations (A2 and A3),

compared to in A1. The difference is particularly pronounced for the rightward directed horizontal flow,

and the downward directed vertical flow, . Regarding and , the results from

A2 and A3 are to a large extent coinciding. In Figure 3-8, development of vertical flow, and horizontal

flow, —extracted from the same stress point in all approaches—are shown. The point is located

approximately 1 m below the initial water level, 1 m to the right from the slope surface.

Figure 3-8: Vertical flow (picture to the left), and horizontal flow (picture to the right); both averages

~0.05 m/day.

Again, larger flow deviations were captured in A2 and A3, compared to in A1.

-60%

-30%

0%

30%

60%

90%

1 2 3 4 5 6 7 8 8 10

|qm

ax|(

dev

iati

on

fro

m

aver

age)

WLFC

-100%

-50%

0%

50%

100%

150%

200%

1 2 3 4 5 6 7 8 9 10qx,

rig

ht (

dev

iati

on

fro

m

aver

age)

WLFC

A1

A2

A3

-100%

-50%

0%

50%

100%

150%

200%

1 2 3 4 5 6 7 8 9 10

qy,

up (

dev

iati

on

fro

m

aver

age)

WLFC

-100%

-50%

0%

50%

100%

150%

200%

1 2 3 4 5 6 7 8 9 10

qy,

do

wn (

dev

iati

on

fro

m

aver

age)

WLFC

A1

A2

A3

-20%

0%

20%

40%

60%

1 2 3 4 5 6 7 8 9 10

qy (

dev

iati

on

fro

m

aver

age)

WLFC

-20%

0%

20%

40%

60%

1 2 3 4 5 6 7 8 9 10

qx (

dev

iati

on

fro

m

aver

age)

WLFC

A1

A2

A3

11

3.3 Parameter influence

The sensitivity analysis did provide parameter-influence values; total sensitivity ( ). In Figure 3-9

the total sensitivity values (on the y-axis) for each of the five parameters at initial state (Init.), after one

rapid drawdown (RDD), and after one WLFC (1), are shown.

(A1) (A2) (A3)

Figure 3-9: The influence (in terms of sensitivity) of each parameter, with respect to safety (in terms of

FOS).

The stiffness parameters are influencing the stability significantly less than are the strength parameters. The

cohesion, exhibits the highest influence in all approaches; followed by the influence of the friction angle,

and the dilatancy angle, . The influence of as well Poisson’s ratio, as of Young’s modulus, is

slightly increasing during further cycling; though remaining below 3% and 2%, respectively. Therefore, the

stiffness parameters are excluded from the presentation of further sensitivity development, presented in

Figure 3-10. ( ) ( ) ( )

Figure 3-10: Sensitivity development. The dotted lines are showing the initial values for each modelling

approach (A10, A20, and A30).

It is clear that the sensitivity values are depending on the stage at which the safety is evaluated, i.e. on the

numbers of WLFC’s run. For cohesion, the qualitative trend is the same in all cases; maximum influence

at drawdown and then declining with increased number of WLFC’s. In A1 the sensitivity of cohesion gets

lower than the initial value after 5 WLFC’s; in A2 this occurs at 2 WLFC’s, and in A3 the initial value is

not reached during the 10 cycles. For the friction angle, the pattern is opposite; minimum values at

drawdown and then increasing. For this parameter, the initial values are exceeded in all approaches; in A1

and A2 at 3 WLFC’s, and in A3 at 6 WLFC’s. The total change (decrease) of the influence of cohesion

0%

20%

40%

60%

80%

ν c φ ψ E

0%

20%

40%

60%

80%

ν c φ ψ E0%

20%

40%

60%

80%

ν c φ ψ E

Init.

RDD

1

50%

52%

54%

56%

58%

60%

1 2 3 4 5 6 7 8 9 10

Sen

siti

vity

WLFC

A30

A10

A20

30%

32%

34%

36%

38%

40%

1 2 3 4 5 6 7 8 9 10WLFC

A1 A2 A3

A30

A10

A20

5%

6%

7%

8%

9%

1 2 3 4 5 6 7 8 9 10

WLFC

A30 A10

A20

12

(during 10 cycles) seems to be similar in all approaches, whereas the increase of the friction-angle influence

in A2 seems to be fading out slower than in A1 and A3. When it comes to the dilatancy angle, the

patterns are more scattered. In all cases the sensitivity values are decreased after the first drawdown, whilst

the further development is almost constant. Though, the absolute magnitudes (and the variations) of the

dilatancy-angle influence are very small, and further conclusions are not reasonably drawn.

4 Discussion

Despite the fact that the absolute magnitudes of differences identified in this study were small—FOS-

values, pore pressures, flows etc.—these do nonetheless demonstrate important dissimilarities concerning

the ability to capture/simulate real soil-water interactions and changes. The results of FOS-development

show that the stability growth continues during the entire period studied; though not smoothly. The stability

increase is to assign to the fact that the groundwater table is dropping (remained at a lower level) for each

WLFC; this was also confirmed by the pore-pressure developments discussed. Lower FOS-values were

obtained using the classical approach, A1 than those obtained using approach A2. This could be reasonably

explained by the consideration taken to the contribution of suction forces to the stability in A2. The fact that

the lowest FOS’s were obtained by using A3, would suggests that this approach is most conservative in this

study, in the sense of not overestimating the stability. This is unexpected since suction was not considered

at all in A1, whereas suction was considered throughout the entire cycle in A3, until the FOS-computation

was run. Anyhow, whether the unexpected result is connected to the nil-steps computed in A3 done in order

to neglected suction, or to unrealistic computation of the soil weight above the water level, the FOS result

of A3 is to be treated with caution.

Use of A1 resulted in smaller vertical crest deformations compared to the other approaches. This

suggests that the two-way interaction between pore pressure changes and deformation development in the

fully coupled computations (A2 and A3), could capture conditions that bring larger deformations. This

means a potential underestimation in A1. Moreover, the flow developments registered in A2 and A3 were

significantly more fluctuating than those in A1; as well horizontally as vertically. The difference was seen

to be particularly pronounced for the rightward directed horizontal flow, and the downward directed

vertical flow. In agreement with what was stated in [31], also the outcomes from the present study show

that there seems to be situations where computations taking into account unsaturated soil behavior, is not

necessarily less conservative than are classical ones.

The parameter-sensitivity analysis results obtained using A1 seems to be quantitatively more

comparable to those obtained using A3 than to those using A2. This suggests that consideration of suction

at stability computation (only true in A2), does directly affect the parameter influence on the results.

However, major differences were not found. Since the sensitivity was computed with respect to FOS

determined by using the strength reduction technique, the low and insignificantly changing sensitivity

values of the stiffness parameters were expected. The non-constant sensitivity values of the dilatancy angle,

might indicate that the occurrences of plastic strains are somewhat different in the different stages. Though,

the influence is significantly lower than for friction angle and cohesion, and also less changing.

It might be useful to consider parameter influence for evaluation of the accuracy needed for definition of

model input data; as well absolute magnitudes, as the development patterns. Moreover, when making up

modelling/design strategies, parameter changes (e.g. by time or by external changes taking place), could be

valuably included.

The results obtained in this study are reflecting effects occurred under the specified conditions of

hydraulic conductivity, rate of water-level change, and slope geometry. Since these factors are

unquestionably affect processes taking place within a watercourse slope, it would be reasonable to include

variation also of such non-constitutive parameters in further studies.

Since fully coupled flow-soil deformation modelling to a higher degree describes real soil-water

interaction, results from semi-coupled approaches (like A1) are probably not optimal for analysis of

processes being governed by fluctuating flows, altering hydraulic gradients, unloading/reloading etc. Since

hydraulic models do partly rely on empirical soil-specific parameters there are potential uncertainties

connected to the level of accuracy of these. Still, this uncertainty is to be related to the extra information

13

possibly obtained at consideration of specific features of unsaturated soils. Flow patterns, changes of

denseness, and non-constant values of the hydraulic conductivity, are all factors being valuable for proper

modelling. Not at least since such changes in turn are governing as well water-transport (i.e. the efficiency

of dissipation of excess pore pressures), as soil-material transport (i.e. susceptibility to internal erosion to

be initiated and/or continued).

5 Concluding remarks

The soil structure seems to become more stable due to the first WLFC, and the stability growth

continues during the entire period studied. This for all approaches.

Modelling assuming strictly saturated or dry conditions (classical, like in A1) resulted in—restricted to

slope stability in terms of safety factors—higher conservatism compared to modelling considering also

the behavior of unsaturated soil and fully hydro-mechanical coupling (advanced, like in A2).

Modelling considering the behavior of unsaturated soil and fully hydro-mechanical coupling, but with

suction neglected at stability computation (A3) resulted in lower safety factors even compared to

classical modelling (A1). This might be explained by errors due to the saturation-suction relation in the

hydraulic model used, or to the nil-step computations run in order to get the negative pore pressure

neglected.

Advanced modelling seems to allow for rapid changes of pore pressures and flow to be more

realistically captured compared to classical modelling. Such changes in turn are governing as well

water-transport (i.e. the efficiency of dissipation of excess pore pressures), as soil-material transport

(i.e. susceptibility to internal erosion to be initiated and/or continued).

The influence of the parameters were changing as a result of WLFC’s taking place; decreased influence

of cohesion, increased influence of the friction angle, scattered patterns of the influence of the

dilatancy angle.

5.1 To be further considered

In order to capture effects of water-level variation frequencies, rates, slope geometries etc., on slope

stability, investigations should be performed using advanced modelling approaches, and considering real

long-term perspectives. Use of a more advanced constitutive soil model (e.g. the Hardening soil model

[32]) would potentially reduce inaccuracy coming from e.g. improper description of plastic deformations

and stiffness changes. The water-level fluctuation effects would also be investigated using models based on

the Basic Barcelona Model [39]. Parameter influence would be preferably considered when evaluating the

accuracy needed for modelling-input data, as well as when making up modelling/design strategies.

Investigations of processes taking place within slopes being subjected to recurrent WLF’s, would desirably

be conducted. For instance by laboratory/scale tests combined with modelling.

6 Acknowledgements

The research has been supported by the Swedish Hydropower Centre, SVC; established by the Swedish

Energy Agency, Elforsk and Svenska Kraftnät together with Luleå University of Technology, The Royal

Institute of Technology, Chalmers University of Technology and Uppsala University. Participating

agencies, companies and industry associations are: Alstom Hydro Sweden, Andritz Hydro,

Energimyndigheten, E.ON Vattenkraft Sverige, Falu Energi och Vatten, Fortum Generation, Holmen

Energi, Jämtkraft, Jönköping Energi, Karlstads Energi, Mälarenergi, Norconsult, Pöyry SwedPower,

Skellefteå Kraft, Sollefteåforsens, Statkraft Sverige, SveMin, Svenska Kraftnät, Sweco Infrastructure,

Sweco Energuide, Umeå Energi, Vattenfall Vattenkraft, VG Power, WSP and ÅF.

14

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Impact ofWater-Level Variations999168/... · 2016-09-30 · LICENTITE TH ESIS Department of Civil, Environmental and Natural Resources Engineering1 Division of Mining and Geotechnical